Math curves have been invented and studied by the world's great mathematicians and scientists and engineers and artists since the days of the ancient Greeks and before.

The construction of a curve, how it may be drawn as the path of a moving point, has been invaluable in the understanding of curves and has led to the invention of new curves.

• The centrepiece is the Astroid - studied by Bernoulli and Leibniz about 1700 but apparently not named until 1836. It is constructed by a wheel rolling inside a circle.
• The 'Archimedes Spiral' was invented by Archimedes about 225BC. It is constructed by the red dot that moves out along the rotating red line.
• The Tautochrone shows that a ball rolling down a Cycloid takes the same time to get to the bottom no matter how high on the side it started. The Cycloid is the curve traced out by a point on the rim of a wheel rolling along a straight line.
• The Catenary is the curve formed by a hanging chain supported at each end (no force on it other than its own weight). The animation shows that its curve (blue) is described by the focus of the rolling red parabola.
• In the 'Four Mice Pursuit', each mouse always runs directly toward the mouse on its right - the resulting path is an 'Equiangular (logarithmic) Spiral' - invented by Descartes in 1638.