some fundamental topics in analytic & euclidean geometry 1
TRANSCRIPT
Microsoft Word - HO Notes 1 Analytic geometry2015Some Fundamental
Topics in Analytic & Euclidean Geometry
1. Cartesian coordinates Analytic geometry, also called coordinate or Cartesian geometry, is the study of geometry using the principles of algebra. The algebra of the real numbers can be employed to yield results about geometry due to the Cantor – Dedekind axiom which asserts that there is a one to one correspondence between the real numbers and the points on a line. Usually the Cartesian coordinate system is applied to manipulate equations for planes, lines, curves, and circles, often in two and sometimes in three dimensions of measurement. It is concerned with defining geometrical shapes in a numerical way and extracting numerical information from that representation. Some consider that the introduction of analytic geometry was the beginning of modern mathematics. History The Greek mathematician Apollonius of Perga, (262-190 BC) in On Determinate Section dealt with problems in a manner that may be called an analytic geometry of one dimension; with the question of finding points on a line that were in a ratio to the others. Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of Descartes by some 1800 years. His application of reference lines, a diameter and a tangent is essentially no different than our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas or x-coordinates, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates or y-coordinates. He further developed relations between
P = (2, –1 .5)
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2. Excerpts from Apollonius, Conics, Book I: FIRST DEFINITIONS 1. If from a point a straight line is joined to the circumference of a circle which is not in the same plane with the point, and the line is produced in both directions, and if, with the point remaining fixed, the straight line being rotated about the circumference of the circle returns to the same place from which it began, then the generated surface composed of the two surfaces lying vertically opposite one another, each of which increases indefinitely as the generating straight line is produced indefinitely, I call a conic surface, and I call the fixed point the vertex, and the straight line drawn from the vertex to the center of the circle I call the axis. 2. And the figure contained by the circle and by the conic surface between the vertex and the circumference of the circle I call a cone, and the point which is also the vertex of the surface I call the vertex of the cone, and the straight line drawn from the vertex to the center of the circle I call the axis, and the circle I call the base of the cone. 3. I call right cones those having axes perpendicular to their bases, and I call oblique those not having axes perpendicular to their bases. 4. Of any curved line which is in one plane, I call that straight line the diameter which, drawn from the curved line, bisects all straight lines drawn to this curved line parallel to some straight line; and I call the end of the diameter situated on the curved line the vertex of the curved line, and I say that each of these parallels is drawn ordinatewise to the diameter. 5. Likewise, of any two curved lines lying in one plane, I call that straight line the transverse diameter which cuts the two curved lines and bisects all the straight lines drawn to either of the curved lines parallel to some straight line; and I call the ends of the [transverse] diameter situated on the curved lines the vertices of the curved lines; and I call that straight line the upright diameter which, lying between the two curved lines, bisects all the straight lines intercepted between the curved lines and drawn parallel to some straight line; and I say that each of
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the parallels is drawn ordinatewise to the [transverse or upright] diameter. 6. The two straight lines, each of which, being a diameter, bisects the straight lines parallel to the other, I call the conjugate diameters of a curved line and of two curved lines. 7. And I call that straight line the axis of a curved line and of two curved lines which being a diameter of the curved line or lines cuts the parallel straight lines at right angles. 8. And I call those straight lines the conjugate axes of a curved line and of two curved lines which, being conjugate diameters, cut the straight lines parallel to each other at right angles. PROPOSITION 1 The straight lines drawn from the vertex of the conic surface to points on the surface are on that surface.
Let there be a conic surface whose vertex is the point A, and let there be taken some point B on the conic surface, and let a straight line ACB be joined. I say that the straight line ACB is on the conic surface. For if possible, let it not be, and let the straight line DE be the line generating the surface, and EF be the circle along which ED is moved. Then if, the point A remaining fixed, the straight line DE is moved along the circumference of the circle EF, it will also go through the point B (Def. 1), and two straight lines will have the same ends. And this is absurd. Therefore the straight line joined from A to B cannot not be on the surface. Therefore it is on the surface. PORISM It is also evident that, if a straight line is joined from the vertex to some point among those within the surface, it will fall within the conic surface; and if it is joined to some point among those without, it will be outside the surface.
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PROPOSITION 2 If on either one of the two vertically opposite surfaces two points are taken, and the straight line joining the points, when produced, does not pass through the vertex, then it will fall within the surface, and produced it will fall outside. Let there be a conic surface whose vertex is the point A, and a circle BC along whose circumference the generating straight line is moved, and let two points D and E be taken on either one of the two vertically opposite surfaces, and let the joining straight line DE, when produced, not pass through the point A. I say that the straight line DE will be within the surface, and produced will be without. Let AE and AD be joined and produced. Then they will fall on the circumference of the circle (I. 1). Let them fall to the points B and C, and let BC be joined. Therefore the straight line BC will be within the circle, and so too within the conic surface. Then let a point F be taken at random on DE, and let the straight line AF be joined and produced. Then it will fall on the straight line BC; for the triangle BCA is in one plane (Eucl. XI. 2). Let it fall to the point G. Since then the point G is within the conic surface, therefore the straight line AG is also within the conic surface (I. 1 porism), and so too the point F is within the conic surface. Then likewise it will be shown that all the points on the straight line DE are within the surface. Therefore the straight line DE is within the surface. Then let DE be produced to H. I say then it will fall outside the conic surface
For if possible, let there be some point H of it not outside the conic surface, and let AH be joined and produced. Then it will fall either on the circumference of the circle or within (I. 1 and porism). And this is impossible, for it falls on BC produced, as for example to the point K. Therefore the straight line EH is outside the surface. Therefore the straight line DE is within the conic surface, and produced is outside.
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Proposition 11 Deriving the Symptom of the Parabola
If a cone is cut by a plane through its axis, and also cut by another plane cutting the base of the cone in a straight line perpendicular to the base of the axial triangle, and if further the diameter of the section is parallel to one side of the axial triangle, then any straight line which is drawn from the section of the cone to its diameter parallel to the common section of the cutting plane and of the cone's base, will equal in square the rectangle contained by the straight line cut off by it on the diameter beginning from the section's vertex and by another straight line which has the ratio to the straight line between the angle of the cone and the vertex of the section that the square on the base of the axial triangle has to the rectangle contained by the remaining two sides of the triangle. And let such a section be called a parabola (παραβολ).
Consider the cone with vertex A and a plane through the axis intersecting the cone. This plane intersects the cone in the axial triangle ABC, where BC is the diameter of the base circle of the cone.
The parabola is like the section of a right-angled cone (orthotome); if the cutting plane is orthogonal to a side of the axial triangle, it must also be parallel to the other side of the axial triangle (for right-angled cones). For Apollonius, the parabola is generated by a plane cutting one side of the axial triangle such that it is parallel to the other side. This works for all cones in general, but we illustrate here with a right circular cone.
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Consider an arbitrary ordinate (i.e. y value) ML constructed on the axis at M. We wish to determine the relationship between ML and EM, that is, the symptom of the conic. The ordinate ML is located in a horizontal plane that cuts the cone in the circle with diameter PR. In this horizontal plane construct the segments PL and LR, which results in a right triangle inscribed in a semicircle. As we have seen before, by similar triangles this implies
or
(1)
Next Apollonius constructs a segment EH perpendicular to EM such that
(2)
How is this possible? All the lengths EA, BC, BA, and AC are known. He is simply finding the point H that makes the ratio true. Now why Apollonius does this is another story. Wait and see.
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Now consider some triangles in the axial plane, namely ABC, APR, and EPM. These triangles are all similar, using the usual properties of parallel lines (see previous derivations for details). From the similarity we have
(3)
and
. (4)
. (5)
Also, in triangle APR, since EM is parallel to AR we know by Elements Book VI, Prop. 2 that
. (6)
. (7)
.
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In this form we can understand Apollonius' use of the word parabola for this section. He has proven that the square on ML is equal to a rectangle applied (paraboli) to a line EH with a width equal to EM. This is based on the idea of application of areas, a Greek technique used to deal with multiplication and division of lengths and areas.
Finally, if we let ML be y, EM be x, and EH be p, we have a standard equation for a parabola,
y2 = p x .
Notice that all that was necessary to derive the symptom of the parabola was a knowledge of similar triangles. We will see this again as we derive the symptom of the next conic section, the ellipse.
Proposition 13 Deriving the Symptom of the Ellipse
If a cone is cut by a plane through its axis, and also cut by another plane on the one hand meeting both sides of the axial triangle, and on the other extended neither parallel to the base nor subcontrariwise, and if the plane the base of the cone is in, and the cutting plane meet in a straight line perpendicular either to the base of the axial triangle or to it produced, then any straight line which is drawn from the section of the cone to the diameter of the section parallel to the common section of the planes, will equal in square some area applied to a straight line to which the diameter of the section has the ratio that the square on the straight line drawn from the cone's vertex to the triangle's base parallel to the sections's diameter has to the rectangle contained by the intercepts of this straight line (on the base) from the sides of the triangle, an area having as breadth the straight line cut off on the diameter beginning from the section's vertex by this straight line from the section to the diameter, and deficient (λλειπου) by a figure similar and similarly situated to the rectangle contained by the diameter and parameter. And let such a section be called an ellipse (λλειψις). We will still consider the cone with vertex A and a plane through the axis intersecting the cone. This plane intersects the cone in the axial triangle ABC, where BC is the diameter of the base circle of the cone.
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To generate the ellipse the cutting plane must intersect both sides of the axial triangle. Once again this works for all cones in general, but we illustrate here with a right circular cone.
Consider an arbitrary ordinate (i.e. y value) ML constructed on the axis at M. We wish to determine the relationship between ML and EM, that is, the symptom of the conic. The ordinate ML is located in a horizontal plane that cuts the cone in the circle with diameter PR. In this horizontal plane construct the segments PL and LR, which results in a right triangle inscribed in a semicircle. As we saw before in the case of the parabola, by similar triangles this implies
or
. (1)
Apollonius next constructs a segment EH, perpendicular to DE, such that
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, (2)
where AK is parallel to ED. [As in the previous section, all the lengths DE, AK, BK, and KC are known. He is simply finding the point H that makes the ratio true.]
Notice in the figure that triangles DEH and DMX are similar. This implies that
(3)
where the first equality is due to the similarity, and the second one is due to the fact that MX = EO. Next, consider the cone diagram carefully (a different viewpoint is below). The triangles ABK, EBG, and EPM in the axial plane are all similar (parallel lines), and also the triangles ACK, DCG, and DMR are all similar (do you see why?). From these two sets of similar triangles we have
(4)
and
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. . (5)
. (6)
Now compare (3) and (6) and notice that they imply PM . MR = EM . EO. However, from (1) we have ML2 = PM . MR , so together we have ML2 = EM . EO.
Taking another look at our ellipse, notice that triangles DEH and XOH are similar. This implies
. (7)
Notice that EO = EH - HO, so ML2 = EM . EO becomes, using (7),
. (8)
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In this form we can understand Apollonius' use of the word ellipse for this section. He has proven that the square on ML is equal to a rectangle applied to a line HE with a width equal to EM and deficient (ellipsis) by a rectangle similar to one contained by ED and HE.
.
Can this be written in the standard form of an ellipse? Complete the square and see.
In the next section we derive the symptom of the hyperbola. The derivation follows that of this section. You will notice very little difference between the two until the very end. In Apollonius's Conics, propositions concerning the ellipse and the hyperbola are frequently proven together, due to their similarities.
Proposition 12 Deriving the Symptom of the Hyperbola
If a cone is cut by a plane through its axis, and also cut by another plane cutting the base of the cone in a straight line perpendicular to the base of the axial triangle, and if the diameter of the section produced meets one side of the axial triangle beyond the vertex of the cone, then any straight line which is drawn from the section to its diameter parallel to the common section of the cutting plane and of the cone's base, will equal in square some area applied to a straight line to which the straight line added along the diameter of the section and subtending the exterior angle of the triangle has a ratio that the square on the straight line drawn from the cone's vertex to the triangle's base parallel to the section's diameter has to the rectangle contained by the sections of the base which this straight line makes when drawn, this area having as breadth the straight line cut off on the diameter beginning from the section's vertex by this straight line from the section to the diameter and exceeding (περβλλου) by a figure (εδος), similar and similarly situated to the rectangle contained by the straight line subtending the exterior angle of the triangle and by the parameter. And let such a section be called an hyperbola (περβολ).
We will once again consider the cone with vertex A and a plane through the axis intersecting the cone. This plane intersects the cone in the axial triangle ABC, where BC is the diameter of the base circle of the cone. To generate the hyperbola the cutting plane must intersect only one side of the axial triangle but also both "top" and "bottom" of the "double cone." As before this works for all cones in general, but we illustrate here with a right circular cone.
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Consider an arbitrary ordinate (i.e. y value) ML constructed on the axis at M. We wish to determine the relationship between ML and EM, that is, the symptom of the conic. The ordinate ML is located in a horizontal plane that cuts the cone in the circle with diameter PR. In this horizontal plane construct the segments PL and LR, which results in a right triangle inscribed in a semicircle. As we saw before in the case of the parabola, by similar triangles this implies
or
. (1)
As with the ellipse, Apollonius next constructs a segment EH, perpendicular to DE, such that
(2)
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where AK is parallel to ED. [As before, all the lengths DE, AK, BK, and KC are known. He is simply finding the point H that makes the ratio true.]
Notice in the figure that triangles DEH and DMX are similar. This implies that
(3)
where the first equality is due to the similarity, and the second one is due to the fact that MX = EO. Next, consider the cone diagram carefully (a different viewpoint is below). The triangles ABK, EBG, and EPM in the axial plane are all similar (parallel lines), and also the triangles ACK, DCG, and DMR are all similar (do you see why?). From these two sets of similar triangles we have
(4)
and
. (5)
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. (6)
Now compare (3) and (6) and notice that they imply PM . MR = EM . EO. However, from (1) we have ML2 = PM . MR, so together we have ML2 = EM . EO.
Taking another look at our hyperbola, notice that triangles DEH and XOH are similar. This implies
. (7)
Here is where the hyperbola derivation changes from that of the ellipse. Notice now that
EO = EH + HO, so ML2 = EM . EO becomes, using (7),
. (8)
From this expression comes Apollonius' use of the word hyperbola for this section. He has proven that the square on ML is equal to a rectangle applied to a line HE with a width equal to EM and exceeding (yperboli) by a rectangle similar to one contained by ED and HE.
.
Can this be written in the standard form of a hyperbola? Again, it is up to you to try it.
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Analytic geometry has traditionally been attributed to René Descartes who made significant progress with the methods of analytic geometry when in 1637 in the appendix entitled Geometry of the titled Discourse on the Method of Rightly Conducting the Reason in the Search for Truth in the Sciences, commonly referred to as Discourse on Method. This work, written in his native French tongue, and its philosophical principles, provided the foundation for calculus in Europe.
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3. Congruence in geometry
Two geometrical objects are called congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object. More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections. Congruent triangles Two triangles are congruent if their corresponding sides are equal in length and their corresponding angles are equal in size.
If triangle ABC is congruent to triangle DEF, the relationship can be written mathematically as: ABC y DEF.
Sufficient evidence for congruence between two triangles in the plane can be shown through the following comparisons:
• SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.
• SSS (Side-Side-Side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.
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• ASA (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. The ASA Postulate was contributed by Thales of Miletus .
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4. Similarity in geometry Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. One can be obtained from the other by uniformly "stretching", possibly with additional rotation, i.e., both have the same shape, or additionally the mirror image is taken, i.e., one has the same shape as the mirror image of the other. For example, all circles are similar to each other, all squares are similar to each other, and all parabolas are similar to each other. On the other hand, ellipses are not all similar to each other, nor are hyperbolas all similar to each other. Two triangles are similar if and only if they have the same three angles, the so-called "AAA" condition.
Similar triangles
If triangle ABC is similar to triangle DEF, then this relation can be denoted as
ABC w DEF
In order for two triangles to be similar, it is sufficient for them to have at least two angles that match. If this is true, then the third angle will also match, since the three angles of a triangle must add up to 180°. This condition guarantees that the side lengths are locked in a common ratio, but can vary proportionally.
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F
Suppose that triangle ABC is similar to triangle DEF in such a way that the angle at vertex A is congruent with the angle at vertex D, the angle at B is congruent with the angle at E, and the angle at C is congruent with the angle at F. Then, once this is known, it is possible to deduce proportionalities between corresponding sides of the two triangles, such as the following:
EF DF
EF BC
DE AB == .
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5. Pythagorean Theorem The Pythagorean theorem:
In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). If we let c be the length of the hypotenuse and a and b be the lengths of the other two sides, the theorem can be expressed as the equation: 222 cba =+
or, solved for c : 22 bac += .
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Proof using similar triangles
Like most of the proofs of the Pythagorean theorem, this one is based on the proportionality of the sides of two similar triangles.
Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. We draw the altitude from point C, and call H its intersection with the side AB. The new triangle ACH is similar to our triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well. By a similar reasoning, the triangle CBH is also similar to ABC. The similarities lead to the two ratios..: As
BC = a, AC = b and AB = c .
so c a =
Summing these two equalities, we obtain
22 ba + = HBc × + AHc × = ) ( AHHBc +×
In other words, the Pythagorean theorem:
222 cba =+
6. Area in geometry
Area is a quantity expressing the size of a figure in the Euclidean plane or on a 2-dimensional surface. Points and lines have zero area, although there are space-filling curves. Depending on the particular definition taken, a figure may have infinite area, for example the entire Euclidean plane. In three dimensions, the analog of area is called a volume.
How to define area Although area seems to be one of the basic notions in geometry, it is not at all easy to define even in the Euclidean plane. Most textbooks avoid defining an area, relying on self-evidence. To make the concept of area meaningful one has to define it, at the very least, on polygons in the Euclidean plane, and it can be done using the following definition:
The area of a polygon in the Euclidean plane is a positive number such that: 1. The area of the unit square is equal to one. 2. Congruent polygons have equal areas. 3. Additivity: If a polygon is a union of two polygons which do not have common
interior points, then its area is the sum of the areas of these polygons. The area of an arbitrary square (i) If n is any positive integer ( i.e. a “whole number” ) then it follows immediately from the additivity property of area that the square with side length n has area: A = 2n square units.
Proof
Observe that there are: 2 . . . nnn
n terms nnn =×=+++ 444 3444 21
unit square tiles which cover the square with side length n units. ∴ A = 2n × (1 2unit ) = 2n square units.
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(ii) If n is any positive integer then it also follows from the additivity property that the square with side length
n 1 units must have area:
A =
Observe that there are 2n square tiles with side lengths
n 1 units which
⇒ A =
1 2n
square units .
( iii ) For any positive integers m and n , the square with side lengths
n m units has area
A = 2
n m square units. We may obtain this result by additivity once again.
Proof In this case, note that there are 2n square tiles of side length
n m units
which cover the square having side length m units. ∴ 2n × A = 2m 2unit
⇒ A = 2
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( iv ) For any positive real number s the area of the square with side length s is A = 2s square units. This follows from the well known property that any positive number s (whether rational or irrational) may be obtained as the limit of a sequence of numbers of the
form i
n m
where im and in are positive integers for all i = 1 , 2 , 3, . . . , Thus,
the difference:
n ms becomes infinitesimally small as i becomes larger and
larger. That is, there exists a sequence:
1
1
n m
→ .
So due to the continuity of geometrical area, it must be the case that
A i =
1. Cartesian coordinates Analytic geometry, also called coordinate or Cartesian geometry, is the study of geometry using the principles of algebra. The algebra of the real numbers can be employed to yield results about geometry due to the Cantor – Dedekind axiom which asserts that there is a one to one correspondence between the real numbers and the points on a line. Usually the Cartesian coordinate system is applied to manipulate equations for planes, lines, curves, and circles, often in two and sometimes in three dimensions of measurement. It is concerned with defining geometrical shapes in a numerical way and extracting numerical information from that representation. Some consider that the introduction of analytic geometry was the beginning of modern mathematics. History The Greek mathematician Apollonius of Perga, (262-190 BC) in On Determinate Section dealt with problems in a manner that may be called an analytic geometry of one dimension; with the question of finding points on a line that were in a ratio to the others. Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of Descartes by some 1800 years. His application of reference lines, a diameter and a tangent is essentially no different than our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas or x-coordinates, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates or y-coordinates. He further developed relations between
P = (2, –1 .5)
2
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2. Excerpts from Apollonius, Conics, Book I: FIRST DEFINITIONS 1. If from a point a straight line is joined to the circumference of a circle which is not in the same plane with the point, and the line is produced in both directions, and if, with the point remaining fixed, the straight line being rotated about the circumference of the circle returns to the same place from which it began, then the generated surface composed of the two surfaces lying vertically opposite one another, each of which increases indefinitely as the generating straight line is produced indefinitely, I call a conic surface, and I call the fixed point the vertex, and the straight line drawn from the vertex to the center of the circle I call the axis. 2. And the figure contained by the circle and by the conic surface between the vertex and the circumference of the circle I call a cone, and the point which is also the vertex of the surface I call the vertex of the cone, and the straight line drawn from the vertex to the center of the circle I call the axis, and the circle I call the base of the cone. 3. I call right cones those having axes perpendicular to their bases, and I call oblique those not having axes perpendicular to their bases. 4. Of any curved line which is in one plane, I call that straight line the diameter which, drawn from the curved line, bisects all straight lines drawn to this curved line parallel to some straight line; and I call the end of the diameter situated on the curved line the vertex of the curved line, and I say that each of these parallels is drawn ordinatewise to the diameter. 5. Likewise, of any two curved lines lying in one plane, I call that straight line the transverse diameter which cuts the two curved lines and bisects all the straight lines drawn to either of the curved lines parallel to some straight line; and I call the ends of the [transverse] diameter situated on the curved lines the vertices of the curved lines; and I call that straight line the upright diameter which, lying between the two curved lines, bisects all the straight lines intercepted between the curved lines and drawn parallel to some straight line; and I say that each of
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the parallels is drawn ordinatewise to the [transverse or upright] diameter. 6. The two straight lines, each of which, being a diameter, bisects the straight lines parallel to the other, I call the conjugate diameters of a curved line and of two curved lines. 7. And I call that straight line the axis of a curved line and of two curved lines which being a diameter of the curved line or lines cuts the parallel straight lines at right angles. 8. And I call those straight lines the conjugate axes of a curved line and of two curved lines which, being conjugate diameters, cut the straight lines parallel to each other at right angles. PROPOSITION 1 The straight lines drawn from the vertex of the conic surface to points on the surface are on that surface.
Let there be a conic surface whose vertex is the point A, and let there be taken some point B on the conic surface, and let a straight line ACB be joined. I say that the straight line ACB is on the conic surface. For if possible, let it not be, and let the straight line DE be the line generating the surface, and EF be the circle along which ED is moved. Then if, the point A remaining fixed, the straight line DE is moved along the circumference of the circle EF, it will also go through the point B (Def. 1), and two straight lines will have the same ends. And this is absurd. Therefore the straight line joined from A to B cannot not be on the surface. Therefore it is on the surface. PORISM It is also evident that, if a straight line is joined from the vertex to some point among those within the surface, it will fall within the conic surface; and if it is joined to some point among those without, it will be outside the surface.
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PROPOSITION 2 If on either one of the two vertically opposite surfaces two points are taken, and the straight line joining the points, when produced, does not pass through the vertex, then it will fall within the surface, and produced it will fall outside. Let there be a conic surface whose vertex is the point A, and a circle BC along whose circumference the generating straight line is moved, and let two points D and E be taken on either one of the two vertically opposite surfaces, and let the joining straight line DE, when produced, not pass through the point A. I say that the straight line DE will be within the surface, and produced will be without. Let AE and AD be joined and produced. Then they will fall on the circumference of the circle (I. 1). Let them fall to the points B and C, and let BC be joined. Therefore the straight line BC will be within the circle, and so too within the conic surface. Then let a point F be taken at random on DE, and let the straight line AF be joined and produced. Then it will fall on the straight line BC; for the triangle BCA is in one plane (Eucl. XI. 2). Let it fall to the point G. Since then the point G is within the conic surface, therefore the straight line AG is also within the conic surface (I. 1 porism), and so too the point F is within the conic surface. Then likewise it will be shown that all the points on the straight line DE are within the surface. Therefore the straight line DE is within the surface. Then let DE be produced to H. I say then it will fall outside the conic surface
For if possible, let there be some point H of it not outside the conic surface, and let AH be joined and produced. Then it will fall either on the circumference of the circle or within (I. 1 and porism). And this is impossible, for it falls on BC produced, as for example to the point K. Therefore the straight line EH is outside the surface. Therefore the straight line DE is within the conic surface, and produced is outside.
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Proposition 11 Deriving the Symptom of the Parabola
If a cone is cut by a plane through its axis, and also cut by another plane cutting the base of the cone in a straight line perpendicular to the base of the axial triangle, and if further the diameter of the section is parallel to one side of the axial triangle, then any straight line which is drawn from the section of the cone to its diameter parallel to the common section of the cutting plane and of the cone's base, will equal in square the rectangle contained by the straight line cut off by it on the diameter beginning from the section's vertex and by another straight line which has the ratio to the straight line between the angle of the cone and the vertex of the section that the square on the base of the axial triangle has to the rectangle contained by the remaining two sides of the triangle. And let such a section be called a parabola (παραβολ).
Consider the cone with vertex A and a plane through the axis intersecting the cone. This plane intersects the cone in the axial triangle ABC, where BC is the diameter of the base circle of the cone.
The parabola is like the section of a right-angled cone (orthotome); if the cutting plane is orthogonal to a side of the axial triangle, it must also be parallel to the other side of the axial triangle (for right-angled cones). For Apollonius, the parabola is generated by a plane cutting one side of the axial triangle such that it is parallel to the other side. This works for all cones in general, but we illustrate here with a right circular cone.
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Consider an arbitrary ordinate (i.e. y value) ML constructed on the axis at M. We wish to determine the relationship between ML and EM, that is, the symptom of the conic. The ordinate ML is located in a horizontal plane that cuts the cone in the circle with diameter PR. In this horizontal plane construct the segments PL and LR, which results in a right triangle inscribed in a semicircle. As we have seen before, by similar triangles this implies
or
(1)
Next Apollonius constructs a segment EH perpendicular to EM such that
(2)
How is this possible? All the lengths EA, BC, BA, and AC are known. He is simply finding the point H that makes the ratio true. Now why Apollonius does this is another story. Wait and see.
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Now consider some triangles in the axial plane, namely ABC, APR, and EPM. These triangles are all similar, using the usual properties of parallel lines (see previous derivations for details). From the similarity we have
(3)
and
. (4)
. (5)
Also, in triangle APR, since EM is parallel to AR we know by Elements Book VI, Prop. 2 that
. (6)
. (7)
.
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In this form we can understand Apollonius' use of the word parabola for this section. He has proven that the square on ML is equal to a rectangle applied (paraboli) to a line EH with a width equal to EM. This is based on the idea of application of areas, a Greek technique used to deal with multiplication and division of lengths and areas.
Finally, if we let ML be y, EM be x, and EH be p, we have a standard equation for a parabola,
y2 = p x .
Notice that all that was necessary to derive the symptom of the parabola was a knowledge of similar triangles. We will see this again as we derive the symptom of the next conic section, the ellipse.
Proposition 13 Deriving the Symptom of the Ellipse
If a cone is cut by a plane through its axis, and also cut by another plane on the one hand meeting both sides of the axial triangle, and on the other extended neither parallel to the base nor subcontrariwise, and if the plane the base of the cone is in, and the cutting plane meet in a straight line perpendicular either to the base of the axial triangle or to it produced, then any straight line which is drawn from the section of the cone to the diameter of the section parallel to the common section of the planes, will equal in square some area applied to a straight line to which the diameter of the section has the ratio that the square on the straight line drawn from the cone's vertex to the triangle's base parallel to the sections's diameter has to the rectangle contained by the intercepts of this straight line (on the base) from the sides of the triangle, an area having as breadth the straight line cut off on the diameter beginning from the section's vertex by this straight line from the section to the diameter, and deficient (λλειπου) by a figure similar and similarly situated to the rectangle contained by the diameter and parameter. And let such a section be called an ellipse (λλειψις). We will still consider the cone with vertex A and a plane through the axis intersecting the cone. This plane intersects the cone in the axial triangle ABC, where BC is the diameter of the base circle of the cone.
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To generate the ellipse the cutting plane must intersect both sides of the axial triangle. Once again this works for all cones in general, but we illustrate here with a right circular cone.
Consider an arbitrary ordinate (i.e. y value) ML constructed on the axis at M. We wish to determine the relationship between ML and EM, that is, the symptom of the conic. The ordinate ML is located in a horizontal plane that cuts the cone in the circle with diameter PR. In this horizontal plane construct the segments PL and LR, which results in a right triangle inscribed in a semicircle. As we saw before in the case of the parabola, by similar triangles this implies
or
. (1)
Apollonius next constructs a segment EH, perpendicular to DE, such that
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, (2)
where AK is parallel to ED. [As in the previous section, all the lengths DE, AK, BK, and KC are known. He is simply finding the point H that makes the ratio true.]
Notice in the figure that triangles DEH and DMX are similar. This implies that
(3)
where the first equality is due to the similarity, and the second one is due to the fact that MX = EO. Next, consider the cone diagram carefully (a different viewpoint is below). The triangles ABK, EBG, and EPM in the axial plane are all similar (parallel lines), and also the triangles ACK, DCG, and DMR are all similar (do you see why?). From these two sets of similar triangles we have
(4)
and
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. . (5)
. (6)
Now compare (3) and (6) and notice that they imply PM . MR = EM . EO. However, from (1) we have ML2 = PM . MR , so together we have ML2 = EM . EO.
Taking another look at our ellipse, notice that triangles DEH and XOH are similar. This implies
. (7)
Notice that EO = EH - HO, so ML2 = EM . EO becomes, using (7),
. (8)
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In this form we can understand Apollonius' use of the word ellipse for this section. He has proven that the square on ML is equal to a rectangle applied to a line HE with a width equal to EM and deficient (ellipsis) by a rectangle similar to one contained by ED and HE.
.
Can this be written in the standard form of an ellipse? Complete the square and see.
In the next section we derive the symptom of the hyperbola. The derivation follows that of this section. You will notice very little difference between the two until the very end. In Apollonius's Conics, propositions concerning the ellipse and the hyperbola are frequently proven together, due to their similarities.
Proposition 12 Deriving the Symptom of the Hyperbola
If a cone is cut by a plane through its axis, and also cut by another plane cutting the base of the cone in a straight line perpendicular to the base of the axial triangle, and if the diameter of the section produced meets one side of the axial triangle beyond the vertex of the cone, then any straight line which is drawn from the section to its diameter parallel to the common section of the cutting plane and of the cone's base, will equal in square some area applied to a straight line to which the straight line added along the diameter of the section and subtending the exterior angle of the triangle has a ratio that the square on the straight line drawn from the cone's vertex to the triangle's base parallel to the section's diameter has to the rectangle contained by the sections of the base which this straight line makes when drawn, this area having as breadth the straight line cut off on the diameter beginning from the section's vertex by this straight line from the section to the diameter and exceeding (περβλλου) by a figure (εδος), similar and similarly situated to the rectangle contained by the straight line subtending the exterior angle of the triangle and by the parameter. And let such a section be called an hyperbola (περβολ).
We will once again consider the cone with vertex A and a plane through the axis intersecting the cone. This plane intersects the cone in the axial triangle ABC, where BC is the diameter of the base circle of the cone. To generate the hyperbola the cutting plane must intersect only one side of the axial triangle but also both "top" and "bottom" of the "double cone." As before this works for all cones in general, but we illustrate here with a right circular cone.
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Consider an arbitrary ordinate (i.e. y value) ML constructed on the axis at M. We wish to determine the relationship between ML and EM, that is, the symptom of the conic. The ordinate ML is located in a horizontal plane that cuts the cone in the circle with diameter PR. In this horizontal plane construct the segments PL and LR, which results in a right triangle inscribed in a semicircle. As we saw before in the case of the parabola, by similar triangles this implies
or
. (1)
As with the ellipse, Apollonius next constructs a segment EH, perpendicular to DE, such that
(2)
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where AK is parallel to ED. [As before, all the lengths DE, AK, BK, and KC are known. He is simply finding the point H that makes the ratio true.]
Notice in the figure that triangles DEH and DMX are similar. This implies that
(3)
where the first equality is due to the similarity, and the second one is due to the fact that MX = EO. Next, consider the cone diagram carefully (a different viewpoint is below). The triangles ABK, EBG, and EPM in the axial plane are all similar (parallel lines), and also the triangles ACK, DCG, and DMR are all similar (do you see why?). From these two sets of similar triangles we have
(4)
and
. (5)
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. (6)
Now compare (3) and (6) and notice that they imply PM . MR = EM . EO. However, from (1) we have ML2 = PM . MR, so together we have ML2 = EM . EO.
Taking another look at our hyperbola, notice that triangles DEH and XOH are similar. This implies
. (7)
Here is where the hyperbola derivation changes from that of the ellipse. Notice now that
EO = EH + HO, so ML2 = EM . EO becomes, using (7),
. (8)
From this expression comes Apollonius' use of the word hyperbola for this section. He has proven that the square on ML is equal to a rectangle applied to a line HE with a width equal to EM and exceeding (yperboli) by a rectangle similar to one contained by ED and HE.
.
Can this be written in the standard form of a hyperbola? Again, it is up to you to try it.
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Analytic geometry has traditionally been attributed to René Descartes who made significant progress with the methods of analytic geometry when in 1637 in the appendix entitled Geometry of the titled Discourse on the Method of Rightly Conducting the Reason in the Search for Truth in the Sciences, commonly referred to as Discourse on Method. This work, written in his native French tongue, and its philosophical principles, provided the foundation for calculus in Europe.
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3. Congruence in geometry
Two geometrical objects are called congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object. More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections. Congruent triangles Two triangles are congruent if their corresponding sides are equal in length and their corresponding angles are equal in size.
If triangle ABC is congruent to triangle DEF, the relationship can be written mathematically as: ABC y DEF.
Sufficient evidence for congruence between two triangles in the plane can be shown through the following comparisons:
• SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.
• SSS (Side-Side-Side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.
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• ASA (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. The ASA Postulate was contributed by Thales of Miletus .
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4. Similarity in geometry Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. One can be obtained from the other by uniformly "stretching", possibly with additional rotation, i.e., both have the same shape, or additionally the mirror image is taken, i.e., one has the same shape as the mirror image of the other. For example, all circles are similar to each other, all squares are similar to each other, and all parabolas are similar to each other. On the other hand, ellipses are not all similar to each other, nor are hyperbolas all similar to each other. Two triangles are similar if and only if they have the same three angles, the so-called "AAA" condition.
Similar triangles
If triangle ABC is similar to triangle DEF, then this relation can be denoted as
ABC w DEF
In order for two triangles to be similar, it is sufficient for them to have at least two angles that match. If this is true, then the third angle will also match, since the three angles of a triangle must add up to 180°. This condition guarantees that the side lengths are locked in a common ratio, but can vary proportionally.
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F
Suppose that triangle ABC is similar to triangle DEF in such a way that the angle at vertex A is congruent with the angle at vertex D, the angle at B is congruent with the angle at E, and the angle at C is congruent with the angle at F. Then, once this is known, it is possible to deduce proportionalities between corresponding sides of the two triangles, such as the following:
EF DF
EF BC
DE AB == .
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5. Pythagorean Theorem The Pythagorean theorem:
In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). If we let c be the length of the hypotenuse and a and b be the lengths of the other two sides, the theorem can be expressed as the equation: 222 cba =+
or, solved for c : 22 bac += .
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Proof using similar triangles
Like most of the proofs of the Pythagorean theorem, this one is based on the proportionality of the sides of two similar triangles.
Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. We draw the altitude from point C, and call H its intersection with the side AB. The new triangle ACH is similar to our triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well. By a similar reasoning, the triangle CBH is also similar to ABC. The similarities lead to the two ratios..: As
BC = a, AC = b and AB = c .
so c a =
Summing these two equalities, we obtain
22 ba + = HBc × + AHc × = ) ( AHHBc +×
In other words, the Pythagorean theorem:
222 cba =+
6. Area in geometry
Area is a quantity expressing the size of a figure in the Euclidean plane or on a 2-dimensional surface. Points and lines have zero area, although there are space-filling curves. Depending on the particular definition taken, a figure may have infinite area, for example the entire Euclidean plane. In three dimensions, the analog of area is called a volume.
How to define area Although area seems to be one of the basic notions in geometry, it is not at all easy to define even in the Euclidean plane. Most textbooks avoid defining an area, relying on self-evidence. To make the concept of area meaningful one has to define it, at the very least, on polygons in the Euclidean plane, and it can be done using the following definition:
The area of a polygon in the Euclidean plane is a positive number such that: 1. The area of the unit square is equal to one. 2. Congruent polygons have equal areas. 3. Additivity: If a polygon is a union of two polygons which do not have common
interior points, then its area is the sum of the areas of these polygons. The area of an arbitrary square (i) If n is any positive integer ( i.e. a “whole number” ) then it follows immediately from the additivity property of area that the square with side length n has area: A = 2n square units.
Proof
Observe that there are: 2 . . . nnn
n terms nnn =×=+++ 444 3444 21
unit square tiles which cover the square with side length n units. ∴ A = 2n × (1 2unit ) = 2n square units.
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(ii) If n is any positive integer then it also follows from the additivity property that the square with side length
n 1 units must have area:
A =
Observe that there are 2n square tiles with side lengths
n 1 units which
⇒ A =
1 2n
square units .
( iii ) For any positive integers m and n , the square with side lengths
n m units has area
A = 2
n m square units. We may obtain this result by additivity once again.
Proof In this case, note that there are 2n square tiles of side length
n m units
which cover the square having side length m units. ∴ 2n × A = 2m 2unit
⇒ A = 2
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( iv ) For any positive real number s the area of the square with side length s is A = 2s square units. This follows from the well known property that any positive number s (whether rational or irrational) may be obtained as the limit of a sequence of numbers of the
form i
n m
where im and in are positive integers for all i = 1 , 2 , 3, . . . , Thus,
the difference:
n ms becomes infinitesimally small as i becomes larger and
larger. That is, there exists a sequence:
1
1
n m
→ .
So due to the continuity of geometrical area, it must be the case that
A i =