(1 + (dy/dx)2)3/2 |d²y/dx²| (1 + (dx/dy)?)3/2 r = |d?x/dy?| r = or The osculating circle (solid red) at a point is the tangent circle that best fits the curve.
Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
Explore the radius of curvature of curves. There can be many circles (hat are tangent to a curve at a particular point, but there is one that provides a "best fit" (Figure). This circle is called an osculating circle of the curve. We define it formally in Section 13.4. The radius of the osculating circle is called the radius of curvature of the curve and can be computed using either of the formulas: Consider the ellipse x 2 + 4 y2 = 16.
(a) Compute the radius of curvature in terms of x and y.
(b) Compute the radius of curvature at (4, 0), (2, -J3), and (0, 2). Sketch the ellipse, plot these three points, and label them with the corresponding radius of curvature.
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