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The tangent plane to a surface at a given point p is defined in an analogous way to the tangent line in the case of curves. It is the best approximation of the surface by a plane at p , and can be obtained as the limiting position of the planes passing through 3 distinct points on the surface close to p as these points converge to p .
k = 1 is the tangent line to the right of the circles looking from c 1 to c 2. k = −1 is the tangent line to the right of the circles looking from c 2 to c 1. The above assumes each circle has positive radius. If r 1 is positive and r 2 negative then c 1 will lie to the left of each line and c 2 to the right, and the two tangent lines will ...
The tangent space of at , denoted by , is then defined as the set of all tangent vectors at ; it does not depend on the choice of coordinate chart :. The tangent space T x M {\displaystyle T_{x}M} and a tangent vector v ∈ T x M {\displaystyle v\in T_{x}M} , along a curve traveling through x ∈ M {\displaystyle x\in M} .
In polar coordinates, the polar tangential angle is defined as the angle between the tangent line to the curve at the given point and ray from the origin to the point. [6] If ψ denotes the polar tangential angle, then ψ = φ − θ, where φ is as above and θ is, as usual, the polar angle.
5.3 Mutually tangent given ... Z = 0 with another line in the projective plane (possibly the line at ... two lines has real coordinates except for ...
Let Xx + Yy + Zz = 0 be the equation of a line, with (X, Y, Z) being designated its line coordinates in a dual projective plane. The condition that the line is tangent to the curve can be expressed in the form F(X, Y, Z) = 0 which is the tangential equation of the curve. At a point (p, q, r) on the curve, the tangent is given by
An osculating curve from a given family of curves is a curve that has the highest possible order of contact with a given curve at a given point; for instance a tangent line is an osculating curve from the family of lines, and has first-order contact with the given curve; an osculating circle is an osculating curve from the family of circles ...
This circle, which is the one among all tangent circles at the given point that approaches the curve most tightly, was named circulus osculans (Latin for "kissing circle") by Leibniz. The center and radius of the osculating circle at a given point are called center of curvature and radius of curvature of the curve at that