![]() Note that a small circle has a large curvature, a large circle has a small curvature, and a straight line has zero curvature (and an infinite radius of curvature).Ĭurvature = 1 / Radius of curvature Radius of curvature = 1 / Curvature To state this another way, if the direction of the curve changed at a rate of one radian per inch (distances being measured along the curve), then the radius of curvature would be one inch. If you use units of radians to measure the angles (one radian = 180 degrees/ Pi), then it turns out that 1 / curvature (that is to say: distance / deltaAngle ) is the radius of curvature, in other words it is the radius of the circle an arc of which would most closely approximate that part of the curve. The rate of this change in direction, per unit length along the curve (deltaAngle / distance) is called the curvature. "Spherical Coordinates."įrom MathWorld-A Wolfram Web Resource.Biology 166 Curvature, contractility, tension and the shaping of surfaces Definition of curvature: A curved line gradually changes direction from one point to the next. Referenced on Wolfram|Alpha Spherical Coordinates Cite this as: Standard Mathematical Tables and Formulae. "Tensor Calculations on Computer: Appendix." Comm. Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Orlando, FL: Academic Press, pp. 102-111, "Spherical Polar Coordinates." §2.5 in Mathematical To Differential Equations and Probability. Apostol,Ģnd ed., Vol. 2: Multi-Variable Calculus and Linear Algebra, with Applications Spherical coordinates of vector (1, 2, 3) ![]() Extreme care is therefore needed when consulting the literature. The following table summarizes a number of conventions The symbol is sometimes also used in place of, instead of, and and instead of. Typically means (radial, azimuthal, polar) to a mathematician but (radial, polar,Īzimuthal) to a physicist. This is especially confusing since the identical Unfortunately, the convention in which the symbols and are reversed (both in meaning and in order listed) is alsoįrequently used, especially in physics. Used in the physics literature is retained (resulting, it is hoped, in a bit lessĬonfusion than a foolish rigorous consistency might engender). Is in spherical harmonics, where the convention The sole exception to this convention in this work Remaining the angle in the - plane and becoming the angle out of that Note that this definition provides a logicalĮxtension of the usual polar coordinates notation, In this work, following the mathematics convention, the symbols for the radial, azimuth, and zenith angleĬoordinates are taken as, , and, respectively. This is the convention commonly used in mathematics. To be distance ( radius) from a point to the origin. Is the latitude) from the positive z-axis Known as the zenith angle and colatitude, When referred to as the longitude), to be the polar angle (also Define to be the azimuthal angle in the - plane from the x-axis That are natural for describing positions on a sphere Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken 1985), are a system of curvilinear coordinates
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