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There are a number of meanings for the word "arc" in mathematics. In general, an arc is any smooth curve joining two points. The length of an arc is known as its arc length. ...

An (ordinary) torus is a surface having genus one, and therefore possessing a single "hole" (left figure). The single-holed "ring" torus is known in older literature as an ...

A parameterization of a minimal surface in terms of two functions f(z) and g(z) as [x(r,phi); y(r,phi); z(r,phi)]=Rint[f(1-g^2); if(1+g^2); 2fg]dz, where z=re^(iphi) and R[z] ...

On the Clebsch diagonal cubic, all 27 of the complex lines present on a general smooth cubic surface are real. In addition, there are 10 points on the surface where three of ...

An area-minimizing surface (rectifiable current) bounded by a smooth curve in R^3 is a smooth submanifold with boundary.

The Clebsch diagonal cubic is a cubic algebraic surface given by the equation x_0^3+x_1^3+x_2^3+x_3^3+x_4^3=0, (1) with the added constraint x_0+x_1+x_2+x_3+x_4=0. (2) The ...

A ruled surface is called a generalized cylinder if it can be parameterized by x(u,v)=vp+y(u), where p is a fixed point. A generalized cylinder is a regular surface wherever ...

A ruled surface is called a generalized cone if it can be parameterized by x(u,v)=p+vy(u), where p is a fixed point which can be regarded as the vertex of the cone. A ...

Let M subset R^3 be a regular surface and u_(p) a unit tangent vector to M, and let Pi(u_(p),N(p)) be the plane determined by u_(p) and the normal to the surface N(p). Then ...

If the Gauss map of a complete minimal surface omits a neighborhood of the sphere, then the surface is a plane. This was proven by Osserman (1959). Xavier (1981) subsequently ...