Ad
related to: conic section equation calculator
Search results
Results From The WOW.Com Content Network
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.
The standard form of the equation of a central conic section is obtained when the conic section is translated and rotated so that its center lies at the center of the coordinate system and its axes coincide with the coordinate axes. This is equivalent to saying that the coordinate system's center is moved and the coordinate axes are rotated to ...
A non-degenerate conic section given by equation can be identified by evaluating . The conic section is: [ 13 ] an ellipse or a circle, if B 2 − 4 A C < 0 {\displaystyle B^{2}-4AC<0} ;
requiring a conic to pass through a point imposes a linear condition on the coordinates: for a fixed (,), the equation + + + + + = is a linear equation in (,,,,,); by dimension counting , five constraints (that the curve passes through five points) are necessary to specify a conic, as each constraint cuts the dimension of possibilities by 1 ...
The equation for a conic section with apex at the origin and tangent to the y axis is + (+) = alternately = + (+) where R is the radius of curvature at x = 0. This formulation is used in geometric optics to specify oblate elliptical ( K > 0 ), spherical ( K = 0 ), prolate elliptical ( 0 > K > −1 ), parabolic ( K = −1 ), and hyperbolic ( K ...
A family of conic sections of varying eccentricity share a focus point and directrix line, including an ellipse (red, e = 1/2), a parabola (green, e = 1), and a hyperbola (blue, e = 2). The conic of eccentricity 0 in this figure is an infinitesimal circle centered at the focus, and the conic of eccentricity ∞ is an infinitesimally separated ...
The problem of intersection of an ellipse/hyperbola/parabola with another conic section leads to a system of quadratic equations, which can be solved in special cases easily by elimination of one coordinate. Special properties of conic sections may be used to obtain a solution. In general the intersection points can be determined by solving the ...
In algebraic geometry, the conic sections in the projective plane form a linear system of dimension five, as one sees by counting the constants in the degree two equations. The condition to pass through a given point P imposes a single linear condition, so that conics C through P form a linear system of dimension 4.