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Saddle surface with normal planes in directions of principal curvatures. In geometry, a normal plane is any plane containing the normal vector of a surface at a particular point. The normal plane also refers to the plane that is perpendicular to the tangent vector of a space curve; (this plane also contains the normal vector) see Frenet ...
where c ∈ ℝ n is the center of the circle (irrelevant since it disappears in the derivatives), a,b ∈ ℝ n are perpendicular vectors of length ρ (that is, a · a = b · b = ρ 2 and a · b = 0), and h : ℝ → ℝ is an arbitrary function which is twice differentiable at t. The relevant derivatives of g work out to be
In Euclidean geometry, for a plane curve C and a given fixed point O, the pedal equation of the curve is a relation between r and p where r is the distance from O to a point on C and p is the perpendicular distance from O to the tangent line to C at the point.
In mathematics, a pedal curve of a given curve results from the orthogonal projection of a fixed point on the tangent lines of this curve. More precisely, for a plane curve C and a given fixed pedal point P , the pedal curve of C is the locus of points X so that the line PX is perpendicular to a tangent T to the curve passing through the point X .
Alternatively, a line can be described as the intersection of two planes. Let L be a line contained in distinct planes a and b with homogeneous coefficients (a 0 : a 1 : a 2 : a 3) and (b 0 : b 1 : b 2 : b 3), respectively. (The first plane equation is =, for example.)
For an observer standing on a plane, all planes parallel to the plane they stand on meet one another at the horizon line. Algebraically, if we take to be the ground, then + will be a plane parallel to the ground (displaced 5 meters from it). These two parallel planes meet one another at the line-at-infinity .
The curvature is taken to be positive if the curve turns in the same direction as the surface's chosen normal, and otherwise negative. The directions in the normal plane where the curvature takes its maximum and minimum values are always perpendicular, if k 1 does not equal k 2, a result of Euler (1760), and are called principal directions.
In hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line (its axis). Given a straight line L and a point P not on L , one can construct a hypercycle by taking all points Q on the same side of L as P , with perpendicular distance to L equal to that ...