The sphere is a surface with constant positive curvature. The gaussian curvature of the sphere is then 1/R² for all the points. If we consider a point P on a sphere of radius R, it is easy to think that all the planes passing by the perpendicular cuts the sphere defining equal curves (circles of radius R). Gauss defined the (gaussian) curvature as the product of those curvatures. Thus, we have to find those planes and measure the radius of the osculating circles. There is a theorem, due to Euler, asserting that the curves with maximum and minimum curvature lie in perpendicular planes. They will cut the bell defining different curves. Then, we shall consider the set of planes passing for that line. Let's consider the flattened bell shown beside: to find the curvature at the apex, first we have to find the line perpendicular to the surface. We will talk about the gaussian curvature, so named in honor of the great mathematician Carl Friedrich Gauss, who gave its description. Let's proceed with our study of the curvature by considering a regular surface. It should be noted that the sign of the curvature in the hyperbola reflects the direction of its concavity. Left: plotted in blue the parabola y=x², right the hyperbola y=1/x. Herewith we show some examples that the students can verify themselves studying the functions. It is true that the radius is a geometrical entity, so it is always positive, but we can interpret a negative value for curvature as a concavity directed bottomwise with respect to our cartesian coordinate system. Its formula requires the first and second derivative of the line in a cartesian coordinate system. The name derives from Circulum osculans because it "kisses" the line around the point P ( osculum = kiss). It is easy to find that circle: it's nothing that the osculating circle. Knowing the circle's radius, we immediately get the curvature of the line around P: it's again k=1/R. We can easily build the circle passing for those three points: it will be a good approximation of the curve as long as goes to zero. A and B can slide symmetrically approaching P between them. Let's consider three points on a curved line: A, B and the middle point P. Happily, it exists an esplicit formula that can provide the radius of curvature. To measure the curvature k we should calculate a limit, but we know that often this is a not-so-easy calculation. This is consistent with our previous thought. If we consider the straight line as a circle whose radius is infinite, we find again that its curvature is zero. The bigger is the radius, the smaller is the curvature. Its curvature is the same for all the points and its value is k=1/R. Let's imagine now the simplest curved line: the circle. It is easy to verify that a straight line has curvature zero. The curvature in a point P is calculated as follows
In other words, it is meaningful to define the curvature for a single point, but it's meaningless to talk about "the curvature of a line". The straightness is defined by the tangent line.īesides, it is easy to find that curvature is a local property, not a global one. The curvature, as suggested by the intuition, is the measure of its deviation from the straightness. Let's imagine a generic line drawn on a plane.
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