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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
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 ...
A curve C containing a point P where the radius of curvature equals r, together with the tangent line and the osculating circle touching C at P. In differential geometry, an osculating curve is a plane curve from a given family that has the highest possible order of contact with another curve.
At such points the second derivative of curvature will be zero. Ccircles which have two-point contact with two points S(t 1), S(t 2) on a curve are bi-tangent circles. The centers of all bi-tangent circles form the symmetry set. The medial axis is a subset of the symmetry set.
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. (this is not the same as saying that f has an extremum).
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 ...
"The spiral itself is not drawn: we see it as the locus of points where the circles are especially close to each other." [1] An osculating circle is a circle that best approximates the curvature of a curve at a specific point. It is tangent to the curve at that point and has the same curvature as the curve at that point. [2]