Explanation Description: Figure 1 shows the center (O) of the circle of curvature at A ( x , y ) lies toward the inner side of the curve. Curvature: Curvature is defined as the rate at which the tangent vector T turns per unit length along the curve is called the curvature. Consider T is the unit tangent vector of a smooth curve, then the curvature of the curve is determined as follows. κ = | d T d s | (1) or κ = 1 | v | | d T d t | (2) Circle of curvature: The circle of curvature is also called as osculating circle. Refer to Figure 1, the circle of curvature at a point A on a plane curve where κ ≠ 0 is the circle in the plane of the curve that, Tangent to the curve at the point A ( x , y ) . The same curvature the curve have at a point A ( x , y ) . The center (O) lies towards the concave or inner side of the curve. Center of curvature: Refer to Figure 1. The centre of curvature of the curve at point A ( x , y ) is the center of the circle of curvature. Radius of curvature: The radius of curvature of the curve at point A ( x , y ) is the radius of the circle of curvature. It is represented by ρ . Radius of curvature = ρ = 1 κ For example: Calculate and graph the osculating circle of the parabola y = x 2 at the origin. Parametrize the parabola using the parameter t = x and is stated by the smooth curve r ( t ) = t i + t 2 j The general expression to calculate the velocity. v = d r d t (3) Here, d r d t is the rate of change in displacement with respect to time. Substitute t i + t 2 j for r ( t ) in equation (3). v = d d t [ t i + t 2 j ] = 1 i + 2 t j The magnitude of v is calculated as follows, | v | = 1 2 + ( 2 t ) 2 = 1 + 4 t 2 The unit tangent vector is, T = v | v | (4) Substitute 1 i + 2 t j for v and 1 + 4 t 2 for | v | in equation (4). T = 1 i + 2 t j 1 + 4 t 2 = 1 1 + 4 t 2 i + 2 t 1 + 4 t 2 j = ( 1 + 4 t 2 ) − 1 2 i + 2 t ( 1 + 4 t 2 ) − 1 2 j Differentiate the above equation with respect to t

Thomas' Calculus: Early Transcendentals (14th Edition)

14th Edition

ISBN: 9780134439020

Author: Joel R. Hass, Christopher E. Heil, Maurice D. Weir

Publisher: PEARSON

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Chapter 13, Problem 8GYR

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State the term curvature, circle of curvature, center of curvature, and radius of curvature for twice-differentiable curves in the plane with an examples.

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Chapter 13, Problem 7GYR

Chapter 13, Problem 9GYR

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Thomas' Calculus: Early Transcendentals (14th Edition)

Ch. 13.1 - In Exercises 1–4, find the given limits. 1. Ch. 13.1 - In Exercises 1–4, find the given limits. 2. Ch. 13.1 - In Exercises 1–4, find the given limits. 3. Ch. 13.1 - In Exercises 1–4, find the given limits. 4. Ch. 13.1 - Motion in the Plane In Exercises 5–8, r(t) is the...Ch. 13.1 - Motion in the Plane In Exercises 5–8, r(t) is the...Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

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