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If the polar line of C with respect to a point Q is a line L, then Q is said to be a pole of L. A given line has (n−1) 2 poles (counting multiplicities etc.) where n is the degree of C. To see this, pick two points P and Q on L. The locus of points whose polar lines pass through P is the first polar of P and this is a curve of degree n−1.
In polar coordinates, the polar tangential angle is defined as the angle between the tangent line to the curve at the given point and ray from the origin to the point. [6] If ψ denotes the polar tangential angle, then ψ = φ − θ , where φ is as above and θ is, as usual, the polar angle.
The equation defining a plane curve expressed in polar coordinates is known as a polar equation. In many cases, such an equation can simply be specified by defining r as a function of φ. The resulting curve then consists of points of the form (r(φ), φ) and can be regarded as the graph of the polar function r.
If a point P moves along a line l, its polar p rotates about the pole L of the line l. If two tangent lines can be drawn from a pole to the conic section, then its polar passes through both tangent points. If a point lies on the conic section, its polar is the tangent through this point to the conic section. If a point P lies on its own polar ...
The tool comes pre-programmed with 36 different example graphs for the purpose of teaching new users about the tool and the mathematics involved. [ 15 ] As of April 2017, Desmos also released a browser-based 2D interactive geometry tool, with supporting features including the plotting of points, lines, circles, and polygons.
Take P to be the origin. For a curve given by the equation F(x, y)=0, if the equation of the tangent line at R=(x 0, y 0) is written in the form + = then the vector (cos α, sin α) is parallel to the segment PX, and the length of PX, which is the distance from the tangent line to the origin, is p.
If the point p lies on the conic Q, the polar line of p is the tangent line to Q at p. The equation, in homogeneous coordinates, of the polar line of the point p with respect to the non-degenerate conic Q is given by = Just as p uniquely determines its polar line (with respect to a given conic), so each line determines a unique pole p.
The distances shown are the radius (OP), polar subtangent (OT), and polar subnormal (ON). The angle θ is the radial angle and the angle ψ of inclination of the tangent to the radius or the polar tangential angle. Let P = (r, θ) be a point on a given curve defined by polar coordinates and let O denote the origin.