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In geometry, an envelope of a planar family of curves is a curve that is tangent to each member of the family at some point, and these points of tangency together form the whole envelope. Classically, a point on the envelope can be thought of as the intersection of two " infinitesimally adjacent" curves, meaning the limit of intersections of ...
No tangent line can be drawn through a point within a circle, since any such line must be a secant line. However, two tangent lines can be drawn to a circle from a point P outside of the circle. The geometrical figure of a circle and both tangent lines likewise has a reflection symmetry about the radial axis joining P to the center point O of ...
The tangent line is said to be "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. The tangent line to a point on a differentiable curve can also be thought of as a tangent line approximation, the graph of the affine function that best approximates the original function at the ...
Supporting lines and tangent lines are not the same thing, [11] but for convex curves, every tangent line is a supporting line. [8] At a point of a curve where a tangent line exists, there can only be one supporting line, the tangent line. [12] Therefore, a smooth curve is convex if it lies on one side of each of its tangent lines.
As an example, count the conic sections tangent to five given lines in the projective plane. [4] The conics constitute a projective space of dimension 5, taking their six coefficients as homogeneous coordinates , and five points determine a conic , if the points are in general linear position , as passing through a given point imposes a linear ...
Instead of passing through points, a different condition on a curve is being tangent to a given line. Being tangent to five given lines also determines a conic, by projective duality, but from the algebraic point of view tangency to a line is a quadratic constraint, so naive dimension counting yields 2 5 = 32 conics tangent to five given lines ...