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So p 1 and p 2 are the roots of the quadratic equation x 2 + x − 1 = 0. The Carlyle circle associated with this quadratic has a diameter with endpoints at (0, 1) and (−1, −1) and center at (−1/2, 0). Carlyle circles are used to construct p 1 and p 2. From the definitions of p 1 and p 2 it also follows that
In mathematics, a quadratic equation (from Latin quadratus 'square') is an equation that can be rearranged in standard form as [1] + + =, where the variable x represents an unknown number, and a, b, and c represent known numbers, where a ≠ 0. (If a = 0 and b ≠ 0 then the equation is linear, not quadratic.)
Because in a continuous function, the function for a sphere is the function for a circle with the radius dependent on z (or whatever the third variable is), it stands to reason that the algorithm for a discrete sphere would also rely on the midpoint circle algorithm. But when looking at a sphere, the integer radius of some adjacent circles is ...
A conic is the curve obtained as the intersection of a plane, called the cutting plane, with the surface of a double cone (a cone with two nappes).It is usually assumed that the cone is a right circular cone for the purpose of easy description, but this is not required; any double cone with some circular cross-section will suffice.
There are several methods for defining quadratic equations for calculating each leg of a Pythagorean triple. [15] A simple method is to modify the standard Euclid equation by adding a variable x to each m and n pair. The m,n pair is treated as a constant while the value of x is varied to produce a "family" of triples based on the selected triple.
Kissing circles. Given three mutually tangent circles (black), there are, in general, two possible answers (red) as to what radius a fourth tangent circle can have. In geometry, Descartes' theorem states that for every four kissing, or mutually tangent, circles, the radii of the circles satisfy a certain quadratic equation. By solving this ...
The signs s 1, s 2 and s 3 on the right-hand sides of the equations may be chosen in eight possible ways, and each choice of signs gives up to two solutions, since the equation for r s is quadratic. This might suggest (incorrectly) that there are up to sixteen solutions of Apollonius' problem.
When the conic section is given in the general quadratic form A x 2 + B x y + C y 2 + D x + E y + F = 0 , {\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0,} the following formula gives the eccentricity e if the conic section is not a parabola (which has eccentricity equal to 1), not a degenerate hyperbola or degenerate ellipse , and not an imaginary ...