Special Curves

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Trisecting an Angle
Special Curves

### Trisection using Special Curves

#### The Cubic Parabola

The idea of using Parabola can be easily extended to "Cubic parabola". The idea is shown in the figure shown below.

##### Formulation of Cubic Parabola

Expanding the idea of the previous method using Parabola, Cubic Parabola can also be used for angle trisection.
Dividing all 3 terms of the Trisection Equation by 2, the result will be

**(1/2)x**^{3} = (3/2)x + a

This suggests that the roots of the trisection equation are the x-values of the intersection of cubic parabola
and a line cutting y-axis at y = a with its slope = 3/2.

********** cubic_parabola_tri_desc.dwg** ********

You can see the process in animation.**animation**

**To create this drawing and animation: **

** Load cubic_parabola.lsp (load "cubic_parabola")**

Then from command line, type **cubic_trisection **

Example: AOB = 60 degrees case

1. Input 2<60. to specify point A

2. Line AO(blue) will cut inner circle at A'.

3. Drop a line from A' perpendiculr to x-axs, and locate point G(green).

4. Find a point H on y-axis such that OG = OH

5. Draw a line through H parallel to a line EF

( y = 3x/2 + 3).

6. This line intersects cubic parabola at point K.

7. Drop a line from point K perpendicular to x-axis.

8. This line intersects the outer circle at point M.

9. angle MOB trisects angle AOB.

Question: How about the other two intersecting points ?

**** cubic_parabola_trisection_60_deg.dwg** **

#### References

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Fun_Math Content Table
Trisecting an Angle
Special Curves

All questions/suggestions should be sent to Takaya Iwamoto

Last Updated Nov 22, 2006

Copyright 2006 Takaya Iwamoto All rights reserved.
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