Mathematical String Art
by
Richard and Donna Goldstein

Sliding Ladder Equation: x^2/3 + y^2/3 = L^2/3 Every line has the same length = L
Parabola rotated 45^o Equation: x² - 2xy + y² - 2Lx - 2Ly + L² = 0 x-intercept: c y-intercept: L - c
Astroid Equation: x^2/3 + y^2/3 = L^2/3 This is a four way sliding ladder

Ellipse Equation: x²/a² + y²/b² = 1
Hyperbola Equation: x²/a² - y²/b² = 1
Octagon All vertices are connected to each other

The mathematics behind these equations requires analytic geometry and calculus. When a family of lines forms another curve it is called the envelope. For more examples and explanations see: Envelopes of Lines and Circles (http://jwilson.coe.uga.edu/Texts.Folder/Envel/Envelopes.html) by J. W. Wilson or Eric W. Weisstein's page Envelope (http://www.astro.virginia.edu/~eww6n/math/Envelope.html). To find the envelope of a curve represented by F(x,y,c) = 0 where c is a parameter repesenting the different members of the family of curves, eliminate c from the simultaneous equations:
F(x,y,c) = 0
F_c(x,y,c) = 0
where the last equation is the partial derivative with respect to c.