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In algebraic geometry an inflection point is defined slightly more generally, as a regular point where the tangent meets the curve to order at least 3, and an undulation point or hyperflex is defined as a point where the tangent meets the curve to order at least 4.
Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot. Tangent plane to a sphere. In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point.
Geometrically, the map from a conic to its dual is one-to-one (since no line is tangent to two points of a conic, as that requires degree 4), and the tangent line varies smoothly (as the curve is convex, so the slope of the tangent line changes monotonically: cusps in the dual require an inflection point in the original curve, which requires ...
If one of the solutions of + + = is also a solution of + + + =, then the corresponding branch of the curve has a point of inflection at the origin. In this case the origin is called a flecnode . If both tangents have this property, so c 0 + 2 m c 1 + m 2 c 2 {\displaystyle c_{0}+2mc_{1}+m^{2}c_{2}} is a factor of d 0 + 3 m d 1 + 3 m 2 d 2 + m 3 ...
The geometric interpretation of an ordinary double point of C * is a line that is tangent to the curve at two points (double tangent) and the geometric interpretation of a cusp of C * is a point of inflection (stationary tangent). Consider for instance, the case of a smooth cubic: =, = =
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.
The general formula for a tangent to a projective curve may apply, but it is worth to make it explicit in this case. Let p = p d + ⋯ + p 0 {\displaystyle p=p_{d}+\cdots +p_{0}} be the decomposition of the polynomial defining the curve into its homogeneous parts, where p i is the sum of the monomials of p of degree i .
In Euclidean plane geometry, a tangent line to a circle is a line that touches the circle at exactly one point, never entering the circle's interior. Tangent lines to circles form the subject of several theorems , and play an important role in many geometrical constructions and proofs .
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