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The black curve has no singularities but has four distinguished points: the two top-most points correspond to the node (double point), as they both have the same tangent line, hence map to the same point in the dual curve, while the two inflection points correspond to the cusps, since the tangent lines first go one way then the other (slope ...
An x value where the y value of the red, or the blue, curve vanishes (becomes 0) gives rise to a local extremum (marked "HP", "TP"), or an inflection point ("WP"), of the black curve, respectively. In geometry , curve sketching (or curve tracing ) are techniques for producing a rough idea of overall shape of a plane curve given its equation ...
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 given point. [3] Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the
The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
Inflection points in differential geometry are the points of the curve where the curvature changes its sign. [2] [3]For example, the graph of the differentiable function has an inflection point at (x, f(x)) if and only if its first derivative f' has an isolated extremum at x.
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: =, = =
Low-order polynomials tend to be smooth and high order polynomial curves tend to be "lumpy". To define this more precisely, the maximum number of inflection points possible in a polynomial curve is n-2, where n is the order of the polynomial equation. An inflection point is a location on the curve where it switches from a positive radius to ...
Two curves in the plane intersecting at a point p are said to have: 0th-order contact if the curves have a simple crossing (not tangent). 1st-order contact if the two curves are tangent. 2nd-order contact if the curvatures of the curves are equal. Such curves are said to be osculating. 3rd-order contact if the derivatives of the curvature are ...