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Dupin's theorem is a tool for determining the curvature lines of a surface by intersection with suitable surfaces (see examples), without time-consuming calculation of derivatives and principal curvatures. The next example shows, that the embedding of a surface into a threefold orthogonal system is not unique. Examples Right circular cone
A conic is the curve obtained as the intersection of a plane, called the cutting plane, with the surface of a double cone (a cone with two nappes).It is usually assumed that the cone is a right circular cone for the purpose of easy description, but this is not required; any double cone with some circular cross-section will suffice.
The slant height of a right circular cone is the distance from any point on the circle of its base to the apex via a line segment along the surface of the cone. It is given by r 2 + h 2 {\displaystyle {\sqrt {r^{2}+h^{2}}}} , where r {\displaystyle r} is the radius of the base and h {\displaystyle h} is the height.
The volume of a cone depends on height and radius. The volume V of a cone depends on the cone's height h and its radius r according to the formula (,) =. The partial derivative of V with respect to r is =,
For instance, every function that is defined on an interval and has a bounded first derivative is Lipschitz continuous. [1] In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem.
The variation formula computations above define the principal symbol of the mapping which sends a pseudo-Riemannian metric to its Riemann tensor, Ricci tensor, or scalar curvature. The principal symbol of the map assigns to each a map from the space of symmetric (0,2)-tensors on to the space of (0,4)-tensors on given by.
In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or combinations thereof. [ 1][ 2][ 3]
Bicone. In geometry, a bicone or dicone (from Latin: bi-, and Greek: di-, both meaning "two") is the three-dimensional surface of revolution of a rhombus around one of its axes of symmetry. Equivalently, a bicone is the surface created by joining two congruent, right, circular cones at their bases. A bicone has circular symmetry and orthogonal ...