# Parametric points along an ellipse equation

• Nov 10th 2006, 08:32 AM
bvdet
Parametric points along an ellipse equation
I am attempting to parametrically layout points along an ellipse in 3D space by dividing a quadrant into equally spaced radial lines from an ellipse center. I have reduced the problem to this formula:
Code:

(x^2/b^2) + ((mx)^2/h^2) - 1 = 0
'b' = the ellipse semi-major axis, 'h' = the semi-minor axis, and 'm' = the tangent of the angle of the radial line. Can someone show me how to reduce this formula to be in terms of 'x'? Once this is solved, I believe I can translate the point locations into 3D space. Thanks.
• Nov 10th 2006, 08:55 AM
Quick
Quote:

Originally Posted by bvdet
I am attempting to parametrically layout points along an ellipse in 3D space by dividing a quadrant into equally spaced radial lines from an ellipse center. I have reduced the problem to this formula:
Code:

(x^2/b^2) + ((mx)^2/h^2) - 1 = 0
'b' = the ellipse semi-major axis, 'h' = the semi-minor axis, and 'm' = the tangent of the angle of the radial line. Can someone show me how to reduce this formula to be in terms of 'x'? Once this is solved, I believe I can translate the point locations into 3D space. Thanks.

I don't know what you're doing, but can't you just solve for x?

$\displaystyle \frac{x^2}{b^2}+\frac{(mx)^2}{h^2}-1=0$

Thus: $\displaystyle \frac{x^2}{b^2}+\frac{m^2x^2}{h^2}=1$

Therefore: $\displaystyle h^2x^2+b^2m^2x^2=b^2h^2$

Then: $\displaystyle \left(h^2+b^2m^2\right)x^2=b^2h^2$

Divide: $\displaystyle x^2=\frac{b^2h^2}{h^2+b^2m^2}$

Finally: $\displaystyle x=\pm \sqrt{\frac{b^2h^2}{h^2+b^2m^2}}$
• Nov 10th 2006, 09:39 AM
bvdet
Quote:

Originally Posted by Quick
I don't know what you're doing, but can't you just solve for x?

$\displaystyle \frac{x^2}{b^2}+\frac{(mx)^2}{h^2}-1=0$

Thus: $\displaystyle \frac{x^2}{b^2}+\frac{m^2x^2}{h^2}=1$

Therefore: $\displaystyle h^2x^2+b^2m^2x^2=b^2h^2$

Then: $\displaystyle \left(h^2+b^2m^2\right)x^2=b^2h^2$

Divide: $\displaystyle x^2=\frac{b^2h^2}{h^2+b^2m^2}$

Finally: $\displaystyle x=\pm \sqrt{\frac{b^2h^2}{h^2+b^2m^2}}$

Thank you! My algebra skills are rusty. The points I am laying out on the ellipse will be used as reference points for structural steel elements in buildings. Typically ellipses are approximated by a series of arcs. For each three reference points I can determine the center point and radius in 3D space for each element. I will be using a Python script in SDS/2 by Design Data.
• Nov 12th 2006, 07:17 AM
bvdet
Below are images of the finished project. Image 1 shows a view of the ellipse generated in 3D space aligned with a structural steel beam. Image 2 is a screenshot in the plane of the ellipse. I created 3 point construction circles parametrically to approximate the ellipse.