Search results
Results from the WOW.Com Content Network
The tangent plane at a regular point is the affine plane in R 3 spanned by these vectors and passing through the point r(u, v) on the surface determined by the parameters. Any tangent vector can be uniquely decomposed into a linear combination of r u {\displaystyle \mathbf {r} _{u}} and r v . {\displaystyle \mathbf {r} _{v}.}
[3] [4] [5] The triangulation starts with a triangulated hexagon at a starting point. This hexagon is then surrounded by new triangles, following given rules, until the surface of consideration is triangulated. If the surface consists of several components, the algorithm has to be started several times using suitable starting points.
Principal curvature directions along with the surface normal, define a 3D orientation frame at a surface point. For example, in case of a cylindrical surface, by physically touching or visually observing, we know that along one specific direction the surface is flat (parallel to the axis of the cylinder) and hence take note of the orientation ...
Ruled surface generated by two Bézier curves as directrices (red, green) A surface in 3-dimensional Euclidean space is called a ruled surface if it is the union of a differentiable one-parameter family of lines. Formally, a ruled surface is a surface in is described by a parametric representation of the form
A saddle point (in red) on the graph of z = x 2 − y 2 (hyperbolic paraboloid). In mathematics, a saddle point or minimax point [1] is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. [2]
A singular point of an implicit surface (in ) is a point of the surface where the implicit equation holds and the three partial derivatives of its defining function are all zero. Therefore, the singular points are the solutions of a system of four equations in three indeterminates.
Gauss's formula shows that the curvature at a point can be calculated as the limit of angle excess α + β + γ − π over area for successively smaller geodesic triangles near the point. Qualitatively a surface is positively or negatively curved according to the sign of the angle excess for arbitrarily small geodesic triangles. [49]
Graphing the set of points (,) in < and < + which satisfy the formula, results in the following plot: [note 1] The formula is a general-purpose method of decoding a bitmap stored in the constant , and it could be used to draw any other image. When applied to the unbounded positive range , the formula tiles a vertical swath of the plane with a ...