Line segments can be straight or not-straight {curve}.
Curves have parts {arc, curve}.
At large positive or negative values, curves can approach straight lines {asymptote, curve}.
Equations with parameters can define sets of curves {family of curves}.
Linkages {Peaucellier's linkage} {Peaucellier linkage} can construct inverse curves.
Figures have length around edges {perimeter, curve}.
Curved line-segment length {rectification of curve}| is integral from x = x1 to x = x2 of (1 + (dy/dx)^2)^0.5 * dx.
Moving curves or surfaces can have contact points or lines with fixed curves or surfaces, with no slippage {rolling motion}. Moving curve rotates around centroid. Centroid revolves around fixed-surface curvature center at contact point or line.
Straight lines can intersect curves to form line segments {secant, curve}|.
Curve parts {segment, curve} lie between two points. Area between arc and chord makes segment. Solid formed by two parallel planes intersecting sphere makes segment.
Normals at curve points can make orthogonal projections {subnormal, curve} onto x-axis.
Tangents at curve points can make orthogonal projections {subtangent} onto x-axis.
Chords {supplemental chord} can go from any circle or ellipse point to diameter end.
Function points {acnode} {isolated point} can have no nearby points that belong to function. Acnodes must be at least double points.
Points {centroid, curve} can be curve-point-coordinate arithmetic means.
Curve branches meet at a point {multiple point}.
Two curve branches can meet at point {node, curve} {double point}. Both branches can have real and distinct tangents {crunode}. Both branches can have real and coincident tangents {cusp, curve}. Both branches can have imaginary tangents at acnodes.
Curves can have double cusps {tacnode} {point of osculation} {osculation point}.
Turning points or maximum or minimum points {stationary point} do not have to be inflection points.
Wires can have a shape {brachistochrone} so that a bead can slide from one end to the other in shortest possible time.
Equation y = a * e^(b*x) defines curve {exponential curve}|.
Points or envelopes can slide along two fixed curves and make curve {glissette}.
Curve families {integral curve} can be differential-equation solutions.
For successive c values, function f(x,y) = c makes curves {isoclinal}. If dy/dx = f(x,y), solutions have graphs.
Lines {Jordan curve} can have no multiple points. Plane closed curves have inside and outside {Jordan curve theorem}.
Two curves {parallel curves} can have same normal, have same curvature center, and be in one-to-one correspondence.
Continuous curves {Peano-Hilbert curve} can fill continuous two-dimensional regions. Start with square. Put four squares inside and connect centers to make polygonal curve. Put four squares inside small squares to make 16 squares and connect centers to make polygonal curve. Continue to make polygonal curve that approaches passing through all points inside starting square.
Hippias of Elias [-430 to -420] invented a curve {quadratrix curve} {trisectrix}. Begin with semicircle and line segment equal to radius but tangent to semicircle at end. Move line segment through semicircle uniformly, keeping direction, until it is tangent to other semicircle end. Simultaneously rotate ray, starting from same semicircle end as line segment, around semicircle until it goes through other semicircle end. Line segment and ray intersect to form a curve.
Curves {quadric curve} can have second-order equations. Surfaces {quadric surface} can have second-order equations.
Continuous curves {Riemann-Weierstrass curve} can have no direction at all points. Start with straight line segment with slope +1 and straight line segment with slope -1, meeting in middle, like this: ^. Then, on line segments, repeat construction, to make shape like M, with four line segments, placing base endpoints half as far apart as before. If repeated, both endpoints approach same point, and slopes approach infinity.
Curves {sine curve} can look like waves.
Curves {sinusoid} {sinusoidal curve} can be like sine curves.
Curves {tangent curve} can have same tangent at points.
Graphs {tangent graph} can look like third-power functions.
x^3 + x * y^2 + a * y^2 - 3 * a * x^2 = 0 defines curves {trisectrix of Maclaurin}, which can trisect angles.
One continuous xy-plane curve {unicursal curve, continuous} can make x and y be finite continuous functions of a parameter.
Curves {envelope, curve}| can be tangential to all curve-family curves. Curve families can have equation f(x,y,a) = 0, for parameter a. Curve families have boundary curves. Eliminating parameter from function and taking partial differential with respect to parameter can find envelope.
Curve normals {evolute} can form envelopes. Curve evolute is curvature-center locus.
Epicycloids {cardioid} can be curves traced by one circle point rolling on fixed equal-radius-circle outside: r = 2 * a * (l - cos(A)), where r is distance from pole, a is fixed-circle radius, pole is where rolling point meets fixed circle, and A is angle to radius. Cardioids have one loop and are special limaçon-curve cases.
Circle points rolling along straight lines make curves {cycloid}. Circle points rolling on fixed-circle outsides make curves {epicycloid}. Circle points rolling on fixed-closed-curve outsides make curves {pericycloid}. Circle points rolling on fixed-circle insides make curves {hypocycloid}. Fixed circles can have four times rolling-circle diameter {astroid}.
Conchoids {limaçon of Pascal} can have circle for fixed curve: r = 2 * a * cos(A) + b, where r is distance from pole, pole is where rolling point meets fixed circle, A is angle to radius, and a and b are constants. If b < 2*a, limaçon of Pascal has two loops. If b > 2*a, limaçon of Pascal has one loop. If b = 2*a, limaçon of Pascal is a cardioid curve.
Curve points or line envelopes rolling along fixed curves make curves {roulette curve}.
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).
Curves {logistic curve} can model growth with growth factors and limiting resources. Growth can depend on growth factors, which can have weights. Limiting resources {bottleneck, growth} can slow growth.
Amount or percentage P at time t is P(t) = A * (1 + m * e^(-t/T)) / (1 + n * e^(-t/T)), where A = constant growth-factor weight, m = original growth-factor amount, n = original limiting-resource amount, and T = time period or inverse growth rate.
Growth is percentage of factor to resource. Denominator and numerator decrease at same rate.
beginning
Original amount depends on relative m and n amounts and on weight A: P(0) = A * (1 + m) / (1 + n). If n is much greater than m and A, denominator is large, and P is zero.
process
At first, growth is exponential with time. Then growth passes through time when growth has constant rate and is linear. Then growth slows exponentially to zero. Amount is then maximum or 100 percent at maturity.
shape
Logistic curve has sigmoid shape.
comparison
Logistic function inverts natural logit function.
For numbers between 0 and 1, functions {logit curve} can be logit(p) = log(p / (1 - p)) = log(p) - log(1 - p). Logarithmic base must be greater than 1, for example, 2 or e. p / (1 - p) is the odds, so logit is logarithm of odds. Logistic function inverts natural logit function. Logit functions can be linear {logit model}: logit(p) = m*x + b. Logit models {logistic regression} can be for linear regression.
Inverse cumulative distribution functions, or normal-distribution quantile functions, depend on error-function inverses {probit curve}. It changes probability into function over real numbers: probit(p) = 2^0.5 * erf^-(2*p - 1). Probit functions can be linear over a large real-number middle range {probit model}.
Logistic curve can look like S {sigmoid curve}| {standard logistic function}. Sigmoid curve starts at minimum or maximum, always increases or decreases, and ends at maximum or minimum. It has one inflection point, near which it grows linearly. It changes exponentially at beginning and end. Amount or percentage P over time t is P(t) = A * (1 + m * e^(-t/T)) / (1 + n * e^(-t/T)). If A = 1, m = 0, n = 1, and T = 1, P(t) = 1 / (1 + e^-t). dP/dt = P * (1 - P), with P(0) = 1/2 and dP(0) / dt = 1/4.
Arctangent, hyperbolic tangent, and error function make sigmoid curves.
Curves {double sigmoid curve} can look like double S. Double sigmoid curves start at minimum or maximum, always increase or decrease, and end at maximum or minimum. They have two inflection points, near which it grows linearly. Growth rate is zero in middle. Double sigmoid curves change exponentially at beginning and end and near middle. Amount or percentage N over variable x is N(x) = (x - d) * (1 - exp(-((x - d) / s)^2)), where d is average, and s is standard deviation.
Logistic curves {Verhulst curve} can have growth rate directly related to current percentage or total amount and directly related to current resource amount. dN / dt = r * N * (1 - N/K), where N is total population, r is growth rate, and K is maximum population possible {carrying capacity, logistic}. Amount or percentage N over time t is N(t) = (K * N(0) * e^(r*t)) / (K + N(0) * (e^(r*t) - 1)), which comes from N(t) = A * (1 + m * e^(-t/T)) / (1 + n * e^(-t/T)), where A ~ 1, m ~ N(0), n ~ N(0) / K, and T = -1/r.
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Date Modified: 2022.0225