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An injective patch is a patch such that x(u_1,v_1)=x(u_2,v_2) implies that u_1=u_2 and v_1=v_2. An example of a patch which is injective but not regular is the function ...

Two figures are said to be similar when all corresponding angles are equal, and are inversely similar when all corresponding angles are equal and described in the opposite ...

A line in the complex plane with slope +/-i. An isotropic line passes through either of the circular points at infinity. Isotropic lines are perpendicular to themselves.

A set of plane measure 0 that contains a circle of every radius.

Let a space curve have line elements ds_N, ds_T, and ds_B along the normal, tangent, and binormal vectors respectively, then ds_N^2=ds_T^2+ds_B^2, (1) where ds_N^2 = ...

The four planes determined by the four altitudes of a tetrahedron and the orthocenters of the corresponding faces pass through the Monge point of the tetrahedron.

Let a patch be given by the map x:U->R^n, where U is an open subset of R^2, or more generally by x:A->R^n, where A is any subset of R^2. Then x(U) (or more generally, x(A)) ...

A line along a normal vector (i.e., perpendicular to some tangent line). If K subset R^d is a centrosymmetric set which has a twice differentiable boundary, then there are ...

The common axis of the three altitude planes of a trihedron.

A point p on a regular surface M in R^3 is said to be parabolic if the Gaussian curvature K(p)=0 but S(p)!=0 (where S is the shape operator), or equivalently, exactly one of ...