![]() When a curved rope is subjected to lateral pressures and also to longitudinal tension, these forces will come into balance where the local pressure, curvature and tension obey the equation: P = C * TĪnd since the radius of curvature is the reciprocal of the curvature: P = T / R Since most biologists are so much more familiar with the alternative field of analytical geometry, based on defining shapes in terms of one or another system of coordinate, we are prone to accept the (very mistaken!) idea that regularity of geometric shape implies the existence of some equivalents of x, y and z coordinates playing a part in the causation of biological shapes. There is an entire field of mathematics, called " differential geometry", that is concerned with defining curves, surfaces and other shapes in terms of their local properties, independent of systems of coordinate axes. Below are shown six different curves all having this exact same variation of curvature as a function of distance along the curve, but starting out at different locations and in different directions. The line starts out with a short straight (zero curvature) region (labeled a), then the curvature rapidly increases in the clockwise direction (in the sections labeled b), then decreases back towards zero (c), passes through zero (at d), then becomes "negative" (in the sense of curving counter-clockwise, in the region labed e), then passes back through zero (at f) to become positive again (g), before becoming very small (in the region labeled h). The drawing above shows a curved line on the left and a graph of its variations in curvature on the right. ![]() The shapes of lines (and also surfaces) can be entirely defined in terms of the variations of their curvature from point to point, independently of any kind of external coordinate system or axes. ![]() When a curve bends back on itself, one says that its curvature has "become negative", or "reversed its sign" but it is arbitrary which part of the curve one says has negative curvature and which part one says has positive curvature. 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. 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.
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