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At each point, the moving line is always tangent to the curve. Its slope is the derivative; green marks positive derivative, red marks negative derivative and black marks zero derivative. The point (x,y) = (0,1) where the tangent intersects the curve, is not a max, or a min, but is a point of inflection. (Note: the figure contains the incorrect ...
The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number. For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more ...
The same reciprocal relation exists between a point P outside the circle and the secant line joining its two points of tangency. If a point P is exterior to a circle with center O, and if the tangent lines from P touch the circle at points T and S, then ∠TPS and ∠TOS are supplementary (sum to 180°). If a chord TM is drawn from the tangency ...
A rising point of inflection is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing. For a smooth curve given by parametric equations , a point is an inflection point if its signed curvature changes from plus to minus or from minus to plus, i.e ...
PC = point of curvature (point at which the curve begins) PT = point of tangent (point at which the curve ends) PI = point of intersection (point at which the two tangents intersect) T = tangent length; C = long chord length (straight line between PC and PT) L = curve length
In this case, the curvature κ is given by φ′(s), where κ is taken to be positive if the curve bends to the left and negative if the curve bends to the right. [1] Conversely, the tangent angle at a given point equals the definite integral of curvature up to that point: [4] [1]
The curve has a unique vertex at the point of tangency with its defining circle. That is, this point is the only point where the curvature reaches a local minimum or local maximum. [ 10 ] The defining circle of the witch is also its osculating circle at the vertex, [ 11 ] the unique circle that "kisses" the curve at that point by sharing the ...
The four-vertex theorem was first proved for convex curves (i.e. curves with strictly positive curvature) in 1909 by Syamadas Mukhopadhyaya. [8] His proof utilizes the fact that a point on the curve is an extremum of the curvature function if and only if the osculating circle at that point has fourth-order contact with the curve; in general the osculating circle has only third-order contact ...