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In geometry, curve sketching (or curve tracing) are techniques for producing a rough idea of overall shape of a plane curve given its equation, without computing the large numbers of points required for a detailed plot. It is an application of the theory of curves to find their main features.
In the theory of quadratic forms, the parabola is the graph of the quadratic form x 2 (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form x 2 + y 2 (or scalings), and the hyperbolic paraboloid is the graph of the indefinite quadratic form x 2 − y 2. Generalizations to more variables yield ...
A parabolic segment is the region bounded by a parabola and line. To find the area of a parabolic segment, Archimedes considers a certain inscribed triangle. The base of this triangle is the given chord of the parabola, and the third vertex is the point on the parabola such that the tangent to the parabola at that point is parallel to the chord.
The universal parabolic constant is the red length divided by the green length. The universal parabolic constant is a mathematical constant.. It is defined as the ratio, for any parabola, of the arc length of the parabolic segment formed by the latus rectum to the focal parameter.
While a parabolic arch may resemble a catenary arch, a parabola is a quadratic function while a catenary is the hyperbolic cosine, cosh(x), a sum of two exponential functions. One parabola is f(x) = x 2 + 3x − 1, and hyperbolic cosine is cosh(x) = e x + e −x / 2 . The curves are unrelated.
A three-dimensional version of parabolic coordinates is obtained by rotating the two-dimensional system about the symmetry axis of the parabolas. Parabolic coordinates have found many applications, e.g., the treatment of the Stark effect and the potential theory of the edges.
Superparabola functions. A superparabola is a geometric curve defined in the Cartesian coordinate system as a set of points (x, y) with = [()], where p, a, and b are positive integers.
The blue parabolas are the corresponding B-spline curve in 3D, consisting of three parabolas. By choosing the NURBS control points and weights, the parabolas are parallel to the opposite face of the gray cone (with its tip at the 3D origin), so dividing by w to project the parabolas onto the w =1 plane results in circular arcs (red circle; see ...