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The witch of Agnesi (curve MP) with labeled points An animation showing the construction of the witch of Agnesi. To construct this curve, start with any two points O and M, and draw a circle with OM as diameter. For any other point A on the circle, let N be the point of intersection of the secant line OA and the tangent line at M.
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020, the company was acquired by American educational technology website Course Hero. [3] [4]
In geometry, an intersection curve is a curve that is common to two geometric objects. In the simplest case, the intersection of two non-parallel planes in Euclidean 3-space is a line . In general, an intersection curve consists of the common points of two transversally intersecting surfaces , meaning that at any common point the surface ...
There will be an intersection if 0 ≤ t ≤ 1 and 0 ≤ u ≤ 1. The intersection point falls within the first line segment if 0 ≤ t ≤ 1, and it falls within the second line segment if 0 ≤ u ≤ 1. These inequalities can be tested without the need for division, allowing rapid determination of the existence of any line segment ...
The second potential problem is that even if the intersection is zero-dimensional, it may be non-transverse, for example, if V is a plane curve and W is one of its tangent lines. The first problem requires the machinery of intersection theory, discussed above in detail, which replaces V and W by more convenient subvarieties using the moving lemma.
These definitions E 1, E 2, and E 3 of the envelope may be different sets. Consider for instance the curve y = x 3 parametrised by γ : R → R 2 where γ(t) = (t,t 3). The one-parameter family of curves will be given by the tangent lines to γ. First we calculate the discriminant . The generating function is
The surfaces q 1 = constant, q 2 = constant, q 3 = constant are called the coordinate surfaces; and the space curves formed by their intersection in pairs are called the coordinate curves. The coordinate axes are determined by the tangents to the coordinate curves at the intersection of three surfaces. They are not in general fixed directions ...
The Möller–Trumbore ray-triangle intersection algorithm, named after its inventors Tomas Möller and Ben Trumbore, is a fast method for calculating the intersection of a ray and a triangle in three dimensions without needing precomputation of the plane equation of the plane containing the triangle. [1]