plane, solid and coordinate geometry conducted by department of mathematics university of moratuwa...

PLANE, SOLID AND COORDINATE GEOMETRY

CONDUCTED BY

DEPARTMENT OF MATHEMATICSUNIVERSITY OF MORATUWA

MS SHANIKA FERDINANDISMR. KEVIN RAJAMOHAN

History and Philosophy of Mathematics

MA0010

Plane Geometry

April 19, 2023Department of Mathematics,UOM

Euclid ( Father of Geometry)

Euclidean GeometryEuclidean geometry is a mathematical system attributed to

the Greek mathematician Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry.

The method consists of assuming a small set of intuitively appealing axioms, and then proving many other propositions (theorems) from those axioms.


Some basic results in Euclidean Geometry

The sum of angles A, B, and C is equal to 180 degrees.

The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).

Thales' theorem: if AC is a diameter then the angle at B is a right angle


Axioms of Euclid’s Geometry

Euclid gives five postulates for plane geometry, stated in terms of constructions:

Let the following be postulated: 1. [It is possible] to draw a straight line from any point to any point. 2. [It is possible] To produce [extend] a finite straight line continuously in a

straight line. 3. [It is possible] To describe a circle with any center and distance [radius]. 4. That all right angles are equal to one another. 5. The parallel postulate: That, if a straight line falling on two straight lines

make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.


Common Notions (Axioms)

1. Things that equal the same thing also equal one another.

2. If equals are added to equals, then the wholes are equal.

3. If equals are subtracted from equals, then the remainders are equal.

4. Things that coincide with one another equal one another.

5. The whole is greater than the part.


Nine point circle

The nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant points, six lying on the triangle itself (unless the triangle is obtuse). They include:

The midpoint of each side of the triangle. The foot of each altitude. The midpoint of the segment of each altitude from its vertex to the

orthocenter (where the three altitudes meet).


Centroid

The centroid (G) of a triangle is the common intersection of the three medians of a triangle. A median of a triangle is the segment from a vertex to the midpoint of the opposite side.


Orthocenter

The orthocenter (H) of a triangle is the common intersection of the three lines containing the altitudes. An altitude is a perpendicular segment from a vertex to the line of the opposite side.


Circumcenter

The circumcenter (C) of a triangle is the point in the plane equidistant from the three vertices of the triangle. Since a point equidistant from two points lies on the perpendicular bisector of the segment determined by two points, (C) is on the perpendicular bisector of each side of the triangle. Note (C) may be outside the triangle.


Euler Line

In geometry, the Euler line, named after Leonhard Euler, is a line determined from any triangle that is not equilateral; it passes through several important points determined from the triangle. In the image, the Euler line is shown in red. It passes through the orthocenter (blue), the circumcenter (green), the centroid (orange), and the center of the nine-point circle (red) of the triangle..


Pythagorean Theorem: Different Proofs

This is a theorem that may have more known proofs than any other; the book Pythagorean Proposition, by Elisha Scott Loomis, contains 367 proofs.

Proof using similar triangles

Let ABC represent a right angle triangle. Draw an altitude from point C and call H its intersection with the side AB. The new triangle ACH is similar to ABC. (by definition of the altitude, they

both have a right angle) Similarly, triangle CBH is similar to ABC.


Proof using similar triangles cont…

The similarities lead to the two ratios:

These can be written as

Summing these two equalities, we obtain

In other words, The Pythagorean theorem:

Exercise: Prove the Pythagorean theorem in one other way.


The Pythagorean Theorem in 3D

The Pythagorean Theorem, which allows you to find the hypotenuse of a right triangle, can also be used in three dimensions to find the diagonal length of a rectangular prism. This is the distance d from one corner of the box to the furthest opposite corner, as shown in the diagram at the right.

The distance can be calculated using:


Polygons

In geometry a polygon is traditionally a plane figure that is bounded by a closed path or circuit, composed of a finite sequence of straight line segments (i.e., by a closed polygonal chain). These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices or corners.

The following are examples of polygons:


Question:

State whether the figure’s below are polygons or not ?

a. b.


Vertex

The vertex of an angle is the point where the two rays that form the angle intersect.

The vertices of a polygon are the points where its sides intersect.


Regular Polygon

A regular polygon is a polygon whose sides are all the same length, and whose angles are all the same. The sum of the angles of a polygon with n sides, where n is 3 or more, is 180° × (n - 2) degrees.


Triangle- Three sided polygon

Equilateral Triangle or Equiangular TriangleA triangle having all three sides of equal length. The angles of an equilateral triangle all measure 60 degrees.

Isosceles TriangleA triangle having two sides of equal length.

Right TriangleA triangle having a right angle. One of the angles of the triangle measures 90 degrees. The side opposite the right angle is called the hypotenuse.


Four sided Polygons

ParallelogramA four-sided polygon with two pairs of parallel sides.

RhombusA four-sided polygon having all four sides of equal length.

TrapezoidA four-sided polygon having exactly one pair of parallel sides. The two sides that are parallel are called the bases of the trapezoid.


Tessellation

A Tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps.

Only three regular polygons tessellate in the Euclidean Plane: Triangles, Squares or Hexagons.

A tessellation of triangles

A tessellation of squares

A tessellation of hexagons


Compass and straightedge

Compass-and-straightedge or ruler-and-compass construction is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass.

Every point constructible using straightedge and compass may be constructed using compass alone. A number of ancient problems in plane geometry impose this restriction.


Trisecting an angle

Angle trisection is the division of an arbitrary angle into three equal angles. It was one of the three geometric problems of antiquity for which solutions using only compass and straightedge were sought. The problem was algebraically proved impossible by Wantzel (1836) French mathematician. Angles may not in general be trisected

The geometric problem of angle trisection can be related to algebra – specifically, the roots of a cubic polynomial – since by the triple-angle formula,


Gauss

Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy and optics. Sometimes known as the “the Prince of Mathematicians" or "the foremost of mathematicians") and "greatest mathematician since antiquity", Gauss had a remarkable influence in many fields of mathematics and science and is ranked as one of history's most influential mathematicians. He referred to mathematics as "the queen of sciences."


CoordinateGeometry


Coordinate Geometry

Cartesian Coordinates

In the two-dimensional Cartesian coordinate system, a point P in the xy-plane is represented by a pair of numbers (x,y).•x is the signed distance from the y-axis to the point P, and •y is the signed distance from the x-axis to the point P.

In the three-dimensional Cartesian coordinate system, a point P in the xyz-space is represented by a triple of numbers (x,y,z).•x is the signed distance from the yz-plane to the point P, •y is the signed distance from the xz-plane to the point P, and •z is the signed distance from the xy-plane to the point P.


Coordinate Geometry

Polar Coordinates The polar coordinate systems are coordinate systems in which a point is

identified by a distance from some fixed feature in space and one or more subtended angles. They are the most common systems of curvilinear coordinates.

The term polar coordinates often refers to circular coordinates (two-dimensional). Other commonly used polar coordinates are cylindrical coordinates and spherical coordinates (both three-dimensional).


Converting Polar and Cartesian coordinates

To convert from Cartesian Coordinates (x,y) to Polar Coordinates (r, ):θ

To convert from Polar coordinates (r, ) to Cartesian θcoordinates


Circle

A circle is the set of points in a plane that are equidistant from a given point . The distance from the center r is called the radius, and the point o is called the center. Twice the radius is known as the diameter .

In Cartesian coordinates, the equation of a circle of radius r centered on (h,k) is


Area of a Circle

This derivation was first recorded by Archimedes in Measurement of a Circle (ca. 225 BC).

If the circle is instead cut into wedges, as the number of wedges increases to infinity, a rectangle results, so


Further Terminology


Ellipse

The ellipse is defined as the locus ( A the set of all points satisfying some condition) of a point (x,y) which moves so that the sum of its distances from two fixed points (called foci, or focuses ) is constant.


Ellipse cont…

Ellipses with Horizontal Major Axis

Ellipses with Vertical Major Axis


Hyperbola

The word "hyperbola" derives from the Greek meaning "over-thrown" or "excessive", from which the English term hyperbole derives. In mathematics a hyperbola is a smooth planar curve having two connected components or branches, each a mirror image of the other and resembling two infinite bows aimed at each other.


Hyperbola cont..

Horizontal transverse axis

Vertical transverse axis


Parabola


A parabola is the set of all points in the plane equidistant from a given line (the conic section directrix) and a given point not on the line (the focus). The focal parameter (i.e., the distance between the directrix and focus) is therefore given by P=2a, where a is the distance from the vertex to the directrix or focus. The surface of revolution obtained by rotating a parabola about its axis of symmetry is called a parabolid.

Spiral

A spiral is typically a planar curve (that is, flat), like the groove on a record or the arms of a spiral galaxy.

A spiral emanates from a central point, getting progressively farther away as it revolves around the point.


Two-dimensional spirals


Cycloid

A cycloid is the locus of a point on the rim of a circle of radius a rolling along a straight line. The cycloid was first studied by Cusa when he was attempting to find the area of a circle by integration. It was studied and named by Galileo in 1599.


Hypocycloid

The path traced out by a point on the edge of a circle of radius b rolling on the outside of a circle of radius a.


Solid Geometry


Sphere

Spherical surface has been defined as the locus of points in three-dimensional space, at a given distance from a given point.

The given point is called the center. The given distance is called a radius.

Sphere is a solid bounded by a spherical surface.


In analytic geometry, a sphere with center (a, b, c) and radius r is the locus of all points (x, y, z) such that

Refer on the Properties of the sphere.The points on the sphere with radius r can be

parameterized by


Ellipsoid

An ellipsoid is a type of quadric surface that is a higher dimensional analogue of an ellipse. The equation of a standard axis-aligned ellipsoid body in an xyz-Cartesian coordinate system is

Where a and b are the equatorial radii (along the x and y axes) and c is the polar radius (along the z-axis).


Hyperboloid

A hyperboloid is a type of surface in three dimensions, described by the equation

Refer the importance of Hyperboloid structures in Construction engineering.


Plot 3d Figures in Matlab


Platonic solids

Tetrahedron, Cube, Octahedron, Dodecahedron & Icosahedron – These 5 solids are called Perfect solids or Platonic solids (in which a constant number of identical regular faces meet at each vertex)

They are known as Perfect, because of their unique construction-They are the only forms we know of, that have multiple sides which all have the same shapes & size.


Archimedean Polyhedra

They are formed from Platonic Solids by cutting off the corners ( Truncated Polyhedra).

It is a solid made out of, more than one polygon.All the vertices are identical.


The 13 Archimedean Solids


Further Topics in

GeometryApril 19, 2023Department of Mathematics,UOM

Euclid’s 5th Postulate

“That if a straight line falling on two straight lines makes the interior angles less that two right angles, the two straight lines, if produced indefinitely , meet on that side on which the angles are less than tow right angles.

In other words: “Through an exterior point of a straight line ( a line not on the straight line) one can construct one and only parallel to the given straight line.

The 5th Postulate is logically consistent in itself and forms an axiomatic system with the other 4 postulates.


But while forming an axiomatic system, the 5th postulate was thought to be dependant on the first 4.

Therefore mathematicians through out the past, redefined the 5th postulate with new theories and gave way to non-Euclidian geometry. E.g. Hyperbolic geometry, Elliptic geometry.


Non-Euclidian Geometry

The axioms of Geometry were formerly regarded as laws of thought which an intelligent mind could neither deny nor investigate.

However, that it is possible to take a set of axioms, wholly or in part contradicting those of Euclid, and build up a Geometry as consistent as his.

Examples of non-Euclidean geometries include the hyperbolic and elliptic geometry, which are contrasted with a Euclidean geometry.

The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines.


Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line:

1. In Euclidean geometry the lines remain at a constant distance from each other, and are known as parallels.

2. In hyperbolic geometry they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultra parallels.

3. In elliptic geometry the lines "curve toward" each other and eventually intersect.


Triangles in different spaces


Hyperbolic Geometry

Hyperbolic geometry (also called Lobachevskian geometry) was created in the first half of the nineteenth century in the midst of attempts to understand Euclid's axiomatic basis for geometry.

It is one type of non-Euclidean geometry that discards Euclid's 5th postulate.

It is replaced by the postulate which states that “Given a line and a point not on it, there is more than one line (infinitely many lines) going through the given point that is parallel to the given line”.


The parallel postulate in Euclidean geometry is equivalent to the statement that, in two dimensional space, for any given line l and point P not on l, there is exactly one line through P that does not intersect l; i.e., that is parallel to l. In hyperbolic geometry there are at least two distinct lines through P which do not intersect l, so the parallel postulate is false.


An example of such a case in hyperbolic geometry , is the hyperbola. Where the hyperbola, though it approaches the asymptote it never meets it.(This violates Euclid’s parallel postulate)

Applications of hyperbolic geometry includes topics such as Toplogy, Group Theory and Complex variables & conformal mapping.


Problems unsolved in geometry.

The Hadwiger problem.The Polygonal illumination problem.The Chromatic Number of the plane.Kissing Numbers.??Perfect cuboids.The Kabon Triangle Problem??There are many more.. Google and explore!!!


Kissing numbers

In d dimensions, the kissing number K(d) is the maximum number of disjoint unit spheres that can touch a given sphere.

What could be K(2) and K(3)??


Kabon triangle problem

The problem is to find how many disjoint triangles can be created with n lines in the plane (K(n))

What could be the sequence of K(n) ??


At the end of this lecture…

We hope you would have been enlightened about the broader perspective of geometry. Namely plane, solid and coordinate geometry.

You would have realized the need for Mathematical thinking and reasoning…!!

We also hope you would take the formulae of different curves in your minds and apply them when you come across mathematical problems.

Please go through any new words you came across..!!


The End


plane, solid and coordinate geometry conducted by department of mathematics university of moratuwa...

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