Appendix B - Circle to Ellipse Transformation Proof 1

Figure 12 - Transformation Starting with circle
All figures shown as part of the circle to ellipse transformation is displayed on a cartesian plane where the center point is (0, 0).

Start with a circle of radius ‘a’, and some arbitrary point (x, yc) as seen in Figure 8 - Transformation Starting with circle. Note that the subscript ‘c’ is used for yc to indicate that this vertical point is for the point on the circle.

Figure 13 - Transformation Step 1

 The first step in the transformation is to rotate the circle on the x axis into the z dimension (depth). The resultant view from the initial perspective is of the ellipse. Note that the vertical height is now designated as b which is the minor axis of the ellipse. Note also that the arbitrary point has moved downward in the vertical direction to the height of y.

When rotating on the x axis, arbitrary point will only have the y coordinate affected and not the x coordinate which will stay the same.

Figure 14 - Transformation Step 2

 The next step is to rotate the original plane on the y axis by 90° clockwise. This rotation creates a side view of the ellipse. Note that the original circle is also shown for reference only.

This last step creates a view where two triangles are superimposed to each other: Triangle with sides ‘abz’, and triangle with sides ‘xyyC ’. These two triangles are similar in that their angles are all the same. Based on the rule of similar triangles, this means that the ratio of any two related segments is also equal. This leads to Equation 7- Similar Triangle Ratios


Equation 7 - Similar Triangle Ratios