in the following slides some geometry basics will be finding ......• start by picking three...

© 2014/EMCEH. Arthaber

In the following slides some geometry basics will be presented:• Finding the center of a circle/arc• Construct a circle from three points• Finding the min/max distance between a point and

circle/arc

The following notation will be used in the explanations:− given objects are marked in GREEN− construction steps are marked in ORANGE− results are marked in RED

(sometimes other colors might also be used for clarity)− pastel colors are used for preceding construction steps


How to find the center of a circle:


How to find the center of a circle/arc:• Start by picking three arbitrary points on the circle or arc!

− To get accurate results it is recommendable to distribute the points uniformly over the circumference of the circle.

− For an arc use the end points and an arbitrary third point in between.


How to find the center of a circle:• Start by picking three arbitrary points on the circle!• Draw with a compass three circles with same radius (the radius must be

sufficient large so that all the circles intersect) around the three points!


How to find the center of a circle:• Start by picking two arbitrary points on the circle!• Draw with a compass three circles!• Draw a line through the intersection points of two previously constructed

circles!


How to find the center of a circle:• Repeat the last step with the remaining intersections!

The intersection of the two lines is the center of the circle. You can increase accuracy by drawing a line through the remaining two intersections.


How to find the center of a circle:• Draw a line through the remaining intersections to increase accuracy!


Construct a circle from three points:

Constructions steps:• First construct the center as in the previous construction.• Afterwards draw the circle


Construct a circle from three points:• Draw with a compass three circles with same radius (the radius must be

sufficient large that all the circles intersect) around the three points!


Construct a circle from three points:• Draw with a compass three circles!• Draw a line through the intersection points of two previously constructed

circles!


Construct a circle from three points:• Repeat the last step with the remaining intersections!

The intersection of the two lines is the center of the searched circle. You can increase accuracy by drawing a line through the remaining two intersections.


Construct a circle from three points:• Draw a line through the remaining intersections to increase accuracy!


Construct a circle from three points:• The intersection of the three lines is the center of the searched circle• Draw the searched circle by using the compass!


Finding the min/max distance between a point and a circle (Ex1):• Example 1: Point is outside the circle


Finding the min/max distance between a point and a circle (Ex1):• Use the compass to draw a circle which intersects with the given circle and

has its center at the given point!


Finding the min/max distance between a point and a circle (Ex1):• Use the compass to draw a circle which intersects with the given circle! • Connect the two intersection points with a line!


Finding the min/max distance between a point and a circle (Ex1):• Use the compass to draw a circle which intersects with the given circle! • Connect the two intersection points with a line!• Construct a line which is orthogonal to the previously drawn line and passes

through A!


Finding the min/max distance between a point and a circle (Ex1):• The intersections between the line and the circle are at the maximum and

minimum distance.

Hint: If the center of the circle is already know, the line constructed in the previous steps can be found easily by drawing a line which goes through the center of the circle and A.


Finding the min/max distance between a point and a circle (Ex2):• Example 2: Point is inside the circle


Finding the min/max distance between a point and a circle (Ex2):• Use the compass to draw a circle which intersects with the given circle and

has its center at the given point!


Finding the min/max distance between a point and a circle (Ex2):• Use the compass to draw a circle which intersects with the given circle! • Connect the two intersection points with a line!• Construct a line which is orthogonal to the previously drawn line and passes

through B!


Finding the min/max distance between a point and a circle (Ex2):• The intersections between the line and the circle are at the maximum and

minimum distance.

Hint: If the center of the circle is already know, the line constructed in the previous steps can be found easily by drawing a line which goes through the center of the circle and A.


Finding the min/max distance between points and an arc (Ex3):


Finding the min/max distance between points and an arc (Ex3):• Draw lines through the center of the arc and the given points! (When the

center point is not known, it must be constructed like slide 3-8)• Intersections with this line are ether maximum or minimum distances. (the

arc could intersect the line twice, once or not)


Finding the min/max distance between points and an arc (Ex3):• Intersections with this line are ether maximum or minimum distances. (the

arc could intersect the line twice, once or not)• The missing points are found at the endpoints of the arc. (Both points have

to be checked)


That’s it, you survived the tutorial!

Questions?Ass.Prof. Dr. Holger ArthaberEmail: [email protected]: CFEG25

mailto:[email protected]

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in the following slides some geometry basics will be finding ......• start by picking three...

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