Geodesic curvature measures how much a curve bends or turns. For example, a straight line has geodesic curvature zero at all of its points. The absolute value of the geodesic curvature of a circle is the same for all its points too and depends only on the circle's radius. A large radius means that the circle is almost straight and so the curvature is small. But a small radius means that the bending is fast and so the curvature is large. The sign of geodesic curvature depends on the direction of the curve. If we are traveling in a circle counterclockwise, the curvature is positive. But if we move in the opposite direction, the curvature is negative.
Suppose we want to find the geodesic curvature of a curve called c at a point called p. First draw the tangent line to the curve at c. One way to draw this line is first, to consider a point p(t) on the curve that is some distance t away from p. Draw the line through p and p(t). Next pick another point (also called p(t)) that is on the curve but closer to p and draw the line through p and the new p(t). If we keep doing this, the lines that we draw should get closer and closer to the tangent line at p. But, it can happen that these lines do not get close to any line at all. In this case the tangent line at p is not defined and neither is the curvature. The curve is said to be non-differentiable if this happens. Suppose now that we have drawn the tangent line. Now draw a circle that is tangent to the tangent line and intersects p. Choose the circle so that the part of the circle close to p is as close as possible to the part of the curve near p. In other words, choose the circle so that it approximates c near p. Then, the absolute value of the geodesic curvature is equal to one divided by the radius of this circle. If we stand at the point p and face in the direction that the curve is going, then the geodesic curvature is positive if the circle that we drew is on our left side. Otherwise, it is negative. The circle that we drew is called the osculating circle of c at p.
Geodesic curvature is defined for curves on surfaces, too. A surface is an object that locally, looks like two-dimensional space. Here are some examples: two-dimensional space, the sphere, the surface of a donut, the surface of a pretzel, a funnel. A geodesic segment in a surface is a curve between two points that is a shortest path between those points. For example, on the sphere a geodesic segment from the north pole to the south pole is an arc of a longitude line. If x and y are two points on a surface S, then the S-distance between them is the length of the shortest path in S between them. For example, the unit sphere-distance between the north and south pole is Pi even though the distance between those points in three-dimensional space is two. A geodesic is a path that is infinite in both directions and has the property that if two points are contained in the geodesic then the part of the geodesic between them is a geodesic segment. On the sphere, geodesics are also called great circles. Geodesics play the same role in the geometry of surfaces as lines do in the geometry of the plane. So, a curve on a surface is differentiable at a point if it has a tangent geodesic at that point. If p is a point on a surface S and r is a positive number, then the set of all points in S that are distance r away from p in S is called the S-circle of radius r centered at p. S-circles play the same role in the geometry of surfaces as circles do in the geometry of the plane. So, the absolute value of the geodesic curvature of a curve on a surface can be defined in the same way as for curves in the plane with the words 'geodesic' and 'S-circle' substituted for 'line' and 'circle'. In particular, the geodesic curvature of a geodesic is always zero. The sign of geodesic curvature on a surface depends on the orientation of the surface. If the surface is not oriented, then the sign is undefined. Otherwise the sign is positive when the ordered pair of vectors (v,w) is positive with respective to the orientation where v points in the direction of the velocity of the curve and w points in the direction of the acceleration of the curve.
Another way to define geodesic curvature uses the derivative (from differential calculus) of a curve and the some vector analysis. If c is a map from [0,1] to the plane, then its derivative at time t is denoted by c'(t). The inner product of two vectors v and w is denoted by <v,w> and equals the cosine of the angle between them multiplied by the product of their lengths. The length of v is denoted by ||v||. The curvature of c at time t is given by the formula k[c](t) = <c''(t), Jc'(t)>/||c'(t)||^3. Here, J is the linear map that sends any point (x,y) to the point (-y, x). If S is a surface inside three-dimensional space, then every point on the surface has a tangent plane. A vector v whose tail is on the surface can be projected on to the tangent plane of the surface to a vector P(v). Then the geodesic curvature of a curve c on S at time t is given by the formula P(c''(t)) = <k[c](t), Jc'(t)>. It can be shown that a curve on a surface is determined by three objects: its starting point, the tangent vector at its starting point, and its geodesic curvature.
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