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Local tangent plane coordinates (LTP) are part of a spatial reference system based on the tangent plane defined by the local vertical direction and the Earth's axis of rotation. They are also known as local ellipsoidal system , [ 1 ] [ 2 ] local geodetic coordinate system , [ 3 ] local vertical, local horizontal coordinates ( LVLH ), or ...
The tangent plane to a surface at a given point p is defined in an analogous way to the tangent line in the case of curves. It is the best approximation of the surface by a plane at p , and can be obtained as the limiting position of the planes passing through 3 distinct points on the surface close to p as these points converge to p .
The desired external tangent lines are the lines perpendicular to these radial lines at those tangent points, which may be constructed as described above. Internal tangents Construction of the inner tangent. A new circle C 3 of radius r 1 + r 2 is drawn centered on O 1. Using the method above, two lines are drawn from O 2 that are tangent to ...
The same holds true for a line and a circle. Two distinct lines cannot be tangent in the plane, although two parallel lines can be considered as tangent at a point at infinity in inversive geometry (see below). [5] [6] The solution circle may be either internally or externally tangent to each of the given circles.
Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle ) is called the reference plane (sometimes fundamental plane ).
Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple. In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates.
This circle, which is the one among all tangent circles at the given point that approaches the curve most tightly, was named circulus osculans (Latin for "kissing circle") by Leibniz. The center and radius of the osculating circle at a given point are called center of curvature and radius of curvature of the curve at that
and the second fundamental form at the origin in the coordinates (x,y) is the quadratic form + +. For a smooth point P on S, one can choose the coordinate system so that the plane z = 0 is tangent to S at P, and define the second fundamental form in the same way.