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The normal section of a surface at a particular point is the curve produced by the intersection of that surface with a normal plane. [1] [2] [3] The curvature of the normal section is called the normal curvature. If the surface is bow or cylinder shaped, the maximum and the minimum of these curvatures are the principal curvatures.
For a plane given by the general form plane equation + + + =, the vector = (,,) is a normal. For a plane whose equation is given in parametric form (,) = + +, where is a point on the plane and , are non-parallel vectors pointing along the plane, a normal to the plane is a vector normal to both and , which can be found as the cross product =.
Distance from the origin O to the line E calculated with the Hesse normal form. Normal vector in red, line in green, point O shown in blue. In analytic geometry, the Hesse normal form (named after Otto Hesse) is an equation used to describe a line in the Euclidean plane, a plane in Euclidean space, or a hyperplane in higher dimensions.
Plane equation in normal form. In Euclidean geometry, a plane is a flat two-dimensional surface that extends indefinitely. Euclidean planes often arise as subspaces of three-dimensional space. A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin.
A plane curve can often be represented in Cartesian coordinates by an implicit equation of the form (,) = for some specific function f.If this equation can be solved explicitly for y or x – that is, rewritten as = or = for specific function g or h – then this provides an alternative, explicit, form of the representation.
Plane equation in normal form. In Euclidean geometry, a plane is a flat two-dimensional surface that extends indefinitely. Euclidean planes often arise as subspaces of three-dimensional space. A prototypical example is one of a room's walls, infinitely extended and assumed infinitesimal thin.
The equation (,,) = can then be thought of as the homogeneous form of (,) = and it defines the same curve when restricted to the Euclidean plane. For example, the homogeneous form of the equation of the line a x + b y + c = 0 {\displaystyle ax+by+c=0} is a x + b y + c z = 0. {\displaystyle ax+by+cz=0.} [ 12 ]
Plane section of an ellipsoid. Earth section paths are plane curves defined by the intersection of an earth ellipsoid and a plane (ellipsoid plane sections).Common examples include the great ellipse (containing the center of the ellipsoid) and normal sections (containing an ellipsoid normal direction).