Spiric Sections

Note

This Mathematica notebook is from the WWW project A Visual Dictionary of Special Plane Curves. Copyright © 1995-2002 by Xah Lee. (http://xahlee.org/)

definition

Spiric Section is the name given to a family of curves formed by the intersection of a torus and a plane that is parallel to the axis of the torus. That is any cross-sections of a donut cut by a knife from top.

equation derivation

The formula of torus with generating circle radius a and center to axis b is

((((Sqrt[(x^2) + (y^2)]) - b))^2) + (z^2) - (a^2) == 0

we want to derive the equation for the curve that is the intersection of this torus with a plane x=c.

eliminate x by replacing it with a we have:

((((Sqrt[(x^2) + (y^2)]) - b))^2) + (z^2) - (a^2) /. x -> c

Expand@%

To ge rid of radical, we move the terms and square both sides

{2 b (Sqrt[(c^2) + (y^2)]), -(a^2) + (b^2) + (c^2) + (y^2) + 
          (z^2)}^2

we have then

4 (b^2) ((c^2) + 
            (y^2)) == (((-(a^2) + (b^2) + (c^2) + (y^2) + (z^2)))^2)

Plot

The family of curves Spiric sections are dependent on two parameters: the shape of the torus, and the cutting plane position. Thus, it cannot be plotted with one singe contour plot.

-(a^2) + (((-b + (Sqrt[(c^2) + (y^2)])))^2) + (z^2) /. {y -> x, 
        z -> y}

Solve[-(a^2) + (((-b + (Sqrt[(c^2) + (x^2)])))^2) + (y^2) == 
        0, c]

{{c -> -Sqrt[a^2 + b^2 - x^2 - y^2 - 2*b*Sqrt[a^2 - y^2]]}, 
  {c -> Sqrt[a^2 + b^2 - x^2 - y^2 - 2*b*Sqrt[a^2 - y^2]]}, 
 {c -> -Sqrt[a^2 + b^2 - x^2 - y^2 + 2*b*Sqrt[a^2 - y^2]]}, 
  {c -> Sqrt[a^2 + b^2 - x^2 - y^2 + 2*b*Sqrt[a^2 - y^2]]}}