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The circular points at infinity, also called the isotropic points, are the pair of (complex) points on the line at infinity through which all circles pass. The circular ...

Given two curves C_1 and C_2 and a fixed point O, let a line from O cut C_1 at Q and C_2 at R. Then the locus of a point P such that OP=QR is the cissoid. The word cissoid ...

All closed surfaces, despite their seemingly diverse forms, are topologically equivalent to spheres with some number of handles or cross-caps. The traditional proof follows ...

A notation used to describe curves. The fundamental principle of Clebsch-Aronhold notation states that if each of a number of forms be replaced by a power of a linear form in ...

A surface which is simultaneously complete and minimal. There have been a large number of fundamental breakthroughs in the study of such surfaces in recent years, and they ...

The geodesics in a complete Riemannian metric go on indefinitely, i.e., each geodesic is isometric to the real line. For example, Euclidean space is complete, but the open ...

Ellipses sharing common foci (left figure). The family of confocal ellipses covers the plane simply, in the sense that there is a unique ellipse passing through each point in ...

Hyperbolas sharing common foci (left figure). The family of confocal hyperbolas covers the plane simply, in the sense that there is a unique hyperbola passing through each ...

A set of quadratic surfaces which share foci. Ellipsoids and one- and two-sheeted hyperboloids can be confocal. These three types of surfaces can be combined to form an ...

Geometric objects lying in a common plane are said to be coplanar. Three noncollinear points determine a plane and so are trivially coplanar. Four points are coplanar iff the ...