non-radial line that crosses the radial line perpendicularly at the point has the equation Otherwise stated is the point in which the tangent intersects the imaginary circle of radius Polar rose A polar rose is a mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation, for any constant γ0 (including 0). If k is an integer, these equations will produce a k-petaled rose if k is odd, or a 2k-petaled rose if k is even. If k is rational, but not an integer, a rose-like shape may form but with overlapping petals. Note that these equations never define a rose with 2, 6, 10, 14, etc. petals. The variable a directly represents the length or amplitude of the petals of the rose, while k relates to their spatial frequency. The constant γ0 can be regarded as a phase angle. Archimedean spiral The Archimedean spiral is a spiral discovered by Archimedes which can also be expressed as a simple polar equation. It is represented by the equation Changing the parameter a will turn the spiral, while b controls the distance between the arms, which for a given spiral is always constant. The Archimedean spiral has two arms, one for and one for . The two arms are smoothly connected at the pole. If , taking the mirror image of one arm across the 90°/270° line will yield the other arm. This curve is notable as one of the first curves, after the conic sections, to be described in a mathematical treatise, and as a prime example of a curve best defined by a polar equation. Conic sections A conic section with one focus on the pole and the other somewhere on the 0° ray (so that the conic's major axis lies along the polar axis) is given by: where e is the eccentricity and is the semi-latus rectum (the perpendicular distance at a focus from the major axis to the curve). If , this equation defines a hyperbola; if , it defines a parabola; and if , it defines an ellipse. The special case of the latter results in a circle of the radius . Intersection of two polar curves The graphs of two polar functions and have possible intersections of three types: In the origin, if the equations and have at least one solution each. All the points where are solutions