Oval points can have distances to two fixed points. Product of distances can be constant {Cassini oval}. Product equals b^2, where b is distance when oval point is equidistant from fixed points.
Secant endpoints can make curves {cissoid}.
Two points on the line passing through fixed point and intersecting fixed curve both maintain equal and constant distance from intersection point and make two curves {conchoid}. Fixed curve is asymptotic to both conchoid branches.
Conchoids {conchoid of Niomedes} can have straight lines as fixed curves. Conchoid of Niomedes can trisect angle and duplicate cube.
Points on flexible but not stretchable thread, kept taut while being wound or unwound on another curve, can trace a curve {involute}. Curve used for winding is traced-curve involute.
For Cassini ovals, distance between both points can be constant {lemniscate curve}.
Feet of perpendiculars from rectangular-hyperbola center to all tangents makes a curve {lemniscate of Bernoulli}. If fixed point is circle radius times square root of two from circle center, and two points are secant-through-fixed-point chord length from fixed point, both point loci make lemniscate of Bernoulli.
Point maintaining same distance to intersection point for all tangents to fixed curve makes curve {orthotomic curve}.
Points where perpendiculars from fixed points meet tangent to fixed curve point {pole, pedal curve} make curves {pedal curve}. For parabolas, pedal curves are tangents at vertices. For ellipses or hyperbolas, pedal curves are auxiliary circles. In pedal curves, perpendicular length relates to length from fixed point to curve point {tangent-polar equation} {pedal equation}.
Point sets {spiral} around a fixed point {center, spiral} can have distance {radius vector, spiral} from center to fixed point that relates to rotation angle {vectorial angle}.
equiangular
Spirals {equiangular spiral} can have equal inclinations of radius vector and tangent vector at all points.
types
Archimedes spiral is r = a*A, where a is constant, A is angle, and r is radius. Spirals {parabolic spiral} {Fermat's spiral} can be r^2 = a*A.
Equiangular spirals {logarithmic spiral} {logistic spiral} can be log(r) = a*A.
reciprocal
Spirals {hyperbolic spiral} can be: r*A = a. Hyperbolic spirals are same as inverse {reciprocal spiral}.
loxodromic spiral
Spirals {loxodromic spiral} {rhumb line, spiral} in spheres can cut meridians at constant angle.
Straight lines can pass through fixed points {pole, strophoid} to intersect all fixed curve points. Two line points maintain same distance to intersection as distance from intersection to another fixed point, not necessarily on line, to make a curve {strophoid}.
Curve points can have distance to a fixed point. Fixed point has distance from coordinate origin. Curves {tractrix} can have constant ratio of distance from curve point to fixed point to distance from origin to fixed point.
For radius-a circle with center at (0,a), a straight line from the origin intersects the circle to define the y-coordinate for all x {witch of Agnesi} {Agnesi witch} {versed sine curve} {versiera} (Maria Agnesi): y = 8 * a^3 / (x^2 + 4 * a^2).
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Date Modified: 2022.0225