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Given two intersecting lines or line segments, the amount of rotation about the point of intersection (the vertex) required to bring one into correspondence with the other is ...

Nielsen's spiral, also called the sici spiral (von Seggern 1993) is the spiral with parametric equations x(t) = aci(t) (1) y(t) = asi(t), (2) where ci(t) is the cosine ...

The path traced out by a fixed point at a radius b>a, where a is the radius of a rolling circle, also sometimes called an extended cycloid. The prolate cycloid contains ...

For a curve with radius vector r(t), the unit tangent vector T^^(t) is defined by T^^(t) = (r^.)/(|r^.|) (1) = (r^.)/(s^.) (2) = (dr)/(ds), (3) where t is a parameterization ...

Let a particle travel a distance s(t) as a function of time t (here, s can be thought of as the arc length of the curve traced out by the particle). The speed (the scalar ...

In the case of a general surface, the distance between two points measured along the surface is known as a geodesic. For example, the shortest distance between two points on ...

A portion of a disk whose upper boundary is a (circular) arc and whose lower boundary is a chord making a central angle theta<pi radians (180 degrees), illustrated above as ...

The Poincaré hyperbolic disk is a two-dimensional space having hyperbolic geometry defined as the disk {x in R^2:|x|<1}, with hyperbolic metric ...

The lituus is an Archimedean spiral with n=-2, having polar equation r^2theta=a^2. (1) Lituus means a "crook," in the sense of a bishop's crosier. The lituus curve originated ...

In Euclidean space R^3, the curve that minimizes the distance between two points is clearly a straight line segment. This can be shown mathematically as follows using ...