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Remark 1: The 2-points–2-tangents property of a parabola is an affine version of the 3-point degeneration of Pascal's theorem. Remark 2: The 2-points–2-tangents property should not be confused with the following property of a parabola, which also deals with 2 points and 2 tangents, but is not related to Pascal's theorem.
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
The graph of a real single-variable quadratic function is a parabola. If a quadratic function is equated with zero, then the result is a quadratic equation . The solutions of a quadratic equation are the zeros (or roots ) of the corresponding quadratic function, of which there can be two, one, or zero.
The involutes of a given curve are a set of parallel curves. For example: the involutes of a circle are parallel spirals (see diagram). And: [17] A parabola has as (two-sided) offsets rational curves of degree 6. A hyperbola or an ellipse has as (two-sided) offsets an algebraic curve of degree 8.
Geometrically, the roots represent the values at which the graph of the quadratic function = + + , a parabola, crosses the -axis: the graph's -intercepts. [3] The quadratic formula can also be used to identify the parabola's axis of symmetry .
The two-dimensional parabolic coordinates form the basis for two sets of three-dimensional orthogonal coordinates. The parabolic cylindrical coordinates are produced by projecting in the -direction. Rotation about the symmetry axis of the parabolae produces a set of confocal paraboloids, the coordinate system of tridimensional parabolic ...
Parabolas have only one focus, so, by convention, confocal parabolas have the same focus and the same axis of symmetry. Consequently, any point not on the axis of symmetry lies on two confocal parabolas which intersect orthogonally (see below). A circle is an ellipse with both foci coinciding at the center.
A parabola, a convex curve that is the graph of the convex function () = In geometry , a convex curve is a plane curve that has a supporting line through each of its points. There are many other equivalent definitions of these curves, going back to Archimedes .
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