The most common statement known as Steiner's theorem (Casey 1893, p. 329) states that the Pascal lines of the hexagons 123456, 143652, and 163254 formed by interchanging the vertices at positions 2, 4, and 6 are concurrent (where the numbers denote the order in which the vertices of the hexagon are taken). The 20 points of concurrence so generated are known as Steiner points.Another theorem due to Steiner lets lines and join a variable point on a conic section to two fixed points on the same conic section. Then and are projectively related.A third "Steiner's theorem" states that if two opposite edges of a tetrahedron move on two fixed skew lines in any way whatsoever but remain fixed in length, then the volume of the tetrahedron remains constant (Altshiller-Court 1979, p. 87)...
The inner and outer spheres tangent internally to a cone and also to a plane intersecting the cone are called Dandelin spheres.The spheres can be used to show that the intersection of the plane with the cone is an ellipse. Let be a plane intersecting a right circular cone with vertex in the curve . Call the spheres tangent to the cone and the plane and , and the circles on which the spheres are tangent to the cone and . Pick a line along the cone which intersects at , at , and at . Call the points on the plane where the sphere are tangent and . Because intersecting tangents have the same length,(1)(2)Therefore,(3)which is a constant independent of , so is an ellipse with .
The term diamond is another word for a rhombus. The term is also used to denote a square tilted at a angle.The diamond shape is a special case of the superellipse with parameter , giving it implicit Cartesian equation(1)Since the diamond is a rhombus with diagonals and , it has inradius(2)(3)Writing as an algebraic curve gives the quartic curve(4)which is a diamond curve with the diamond edges extended to infinity.When considered as a polyomino, the diamond of order can be considered as the set of squares whose centers satisfy the inequality . There are then squares in the order- diamond, which is precisely the centered square number of order . For , 2, ..., the first few values are 1, 5, 13, 25, 41, 61, 85, 113, 145, ... (OEIS A001844).The diamond is also the name given to the unique 2-polyiamond...
If three conics pass through two given points and , then the lines joining the other two intersections of each pair of conics are concurrent at a point (Evelyn 1974, p. 15). The converse states that if two conics and meet at four points , , , and , and if and are chords of and , respectively, which meet on , then the six points lie on a conic. The dual of the theorem states that if three conics share two common tangents, then their remaining pairs of common tangents intersect at three collinear points.If the points and are taken as the points at infinity, then the theorem reduces to the theorem that radical lines of three circles are concurrent in a point known as the radical center (Evelyn 1974, p. 15).If two of the points and are taken as the points at infinity, then the theorem becomes that if two circles and pass through two points and on a conic , then the lines determined by the pair of intersections of each circle with the conic are parallel (Evelyn..
A toric section is a curve obtained by slicing a torus (generally a horn torus) with a plane. A spiric section is a special case of a toric section in which the slicing plane is perpendicular to both the midplane of the torus and to the plane .Consider a torus with tube radius . For a cutting plane parallel to the -plane, the toric section is either a single circle (for ) or two concentric circles (for ). For planes containing the z-axis, the section is two equal circles.Toric sections at oblique angles can be more complicated, passing from a crescent shape, through a U-shape, and into two disconnected kidney-shaped curves.
A cylindric section is the intersection of a plane with a right circular cylinder. It is a circle (if the plane is at a right angle to the axis), an ellipse, or, if the plane is parallel to the axis, a single line (if the plane is tangent to the cylinder), pair of parallel lines bounding an infinite rectangle (if the plane cuts the cylinder), or no intersection at all (if the plane misses the cylinder entirely; Hilbert and Cohn-Vossen 1999, pp. 7-8).
A curve on a surface whose tangents are always in the direction of principalcurvature. The equation of the lines of curvature can be writtenwhere and are the coefficients of the first and second fundamental forms.
The envelope of the lines connecting corresponding points on the Jacobian curve and Steinerian curve. The Cayleyian curve of a net of curves of order has the same curve genus as the Jacobian curve and Steinerian curve and, in general, the class .
The bifolium is a folium with . The bifolium is a quartic curve and is given by the implicit equation is(1)and the polar equation(2)The bifolium has area(3)(4)(5)Its arc length is(6)(7)(OEIS A118307), where , , , and are elliptic integrals with(8)(9)The curvature is given by(10)(11)The bifolium is the pedal curve of the deltoid where the pedal point is the midpoint of one of the three curved sides.
The necessary and sufficient condition that an algebraic curve has an algebraic involute is that the arc length is a two-valued algebraic function of the coordinates of the extremities. Furthermore, this function is a root of a quadratic equation whose coefficients are rational functions of and .
Guilloché patterns are spirograph-like curves that frame a curve within an inner and outer envelope curve. They are used on banknotes, securities, and passports worldwide for added security against counterfeiting. For currency, the precise techniques used by the governments of Russia, the United States, Brazil, the European Union, Madagascar, Egypt, and all other countries are likely quite different. The figures above show the same guilloche pattern plotted in polar and Cartesian coordinates generated by a series of nested additions and multiplications of sinusoids of various periods.Guilloché machines (alternately called geometric lathes, rose machines, engine-turners, and cycloidal engines) were first used for a watch casing dated 1624, and consist of myriad gears and settings that can produce many different patterns. Many goldsmiths, including Fabergè, employed guilloché machines.The..
The radial curve of a unit circle from a radial point and parametric equations(1)(2)is another circle with parametricequations(3)(4)
The pedal curve of a unitcircle with parametric equation(1)(2)with pedal point is(3)(4)The pedal curve with respect to the center is thecircle itself (Gray 1997, pp. 119 and 124-135).If the pedal point is taken on the circumference (and in particular at the point ), the pedal curve is the cardioid(5)(6)and otherwise is a limaçon.
For a unit circle with parametricequations(1)(2)the negative pedal curve with respect to the pedal point is(3)(4)Therefore if the point is inside the circle (), the negative pedal is an ellipse, if , it is a single point, if the point is outside the circle (), the negative pedal is a hyperbola.
The pedal curve of circleinvolute(1)(2)with the center as the pedal point is the Archimedes'spiral(3)(4)
Consider a unit circle and a radiant point located at . There are four different regimes of caustics, illustrated above.For radiant point at , the catacaustic is the nephroid(1)(2)(Trott 2004, p. 17, mistakenly states that the catacaustic for parallel light falling on any concave mirror is a nephroid.)For radiant point a finite distance , the catacaustic is the curve(3)(4)which is apparently incorrectly described as a limaçonby Lawrence (1972, p. 207).For radiant point on the circumference of the circle (), the catacaustic is the cardioid(5)(6)with Cartesian equation(7)For radiant point inside the circle, the catacausticis a discontinuous two-part curve.If the radiant point is the origin, then the catacaustic degenerates to a single point at the origin since all rays reflect upon themselves back through the origin...
A circle is the set of points in a plane that are equidistant from a given point . The distance from the center is called the radius, and the point is called the center. Twice the radius is known as the diameter . The angle a circle subtends from its center is a full angle, equal to or radians.A circle has the maximum possible area for a given perimeter,and the minimum possible perimeter for a given area.The perimeter of a circle is called the circumference, and is given by(1)This can be computed using calculus using the formula for arc length in polar coordinates,(2)but since , this becomes simply(3)The circumference-to-diameter ratio for a circle is constant as the size of the circle is changed (as it must be since scaling a plane figure by a factor increases its perimeter by ), and also scales by . This ratio is denoted (pi), and has been proved transcendental.Knowing , the area of the circle can be computed either geometrically or using calculus. As the..
Given a parabola with parametricequations(1)(2)the negative pedal curve for a pedal point has equation(3)(4)Taking the pedal point at the origin gives(5)(6)which is a semicubical parabola. Similarly, taking gives(7)(8)which is a Tschirnhausen cubic.
Given a rectangle having sides in the ratio , the golden ratio is defined such that partitioning the original rectangle into a square and new rectangle results in a new rectangle having sides with a ratio . Such a rectangle is called a golden rectangle. Euclid used the following construction to construct them. Draw the square , call the midpoint of , so that . Now draw the segment , which has length(1)and construct with this length. Now complete the rectangle , which is golden since(2)Successive points dividing a golden rectangle into squares lie on a logarithmic spiral (Wells 1991, p. 39; Livio 2002, p. 119) which is sometimes known as the golden spiral.The spiral is not actually tangent at these points, however, but passes through them and intersects the adjacent side, as illustrated above.If the top left corner of the original square is positioned at (0, 0), the center of the spiral occurs at the position(3)(4)(5)(6)(7)(8)(9)(10)(11)and..
A curve of constant width constructed by drawing arcs from each polygon vertex of an equilateral triangle between the other two vertices. The Reuleaux triangle has the smallest area for a given width of any curve of constant width. Let the arc radius be . Since the area of each meniscus-shaped portion of the Reuleaux triangle is a circular segment with opening angle ,(1)(2)But the area of the central equilateral triangle with is(3)so the total area is then(4)Because it can be rotated inside a square, as illustratedabove, it is the basis for the Harry Watt square drill bit.When rotated inside a square of side length 2 having corners at ), the envelope of the Reuleaux triangle is a region of the square with rounded corners. At the corner , the envelope of the boundary is given by the segment of the ellipse with parametric equations(5)(6)for , extending a distance from the corner (Gleißner and Zeitler 2000). The ellipse has center , semimajor axis..
There are at least two theorems known as Salmon's theorem. This first states that if and are two points, and are the perpendiculars from and to the polars of and , respectively, with respect to a circle with center , then (Durell 1928; Salmon 1954, §101, p. 93).The second Salmon's theorem states that, given a track bounded by two confocal ellipses, if a ball is rolled so that its trajectory is tangent to the inner ellipse, the ball's trajectory will be tangent to the inner ellipse following all subsequent caroms as well (Salmon 1954, §189, pp. 181-182).
A fast boat is overtaking a slower one when fog suddenly sets in. At this point, the boat being pursued changes course, but not speed, and proceeds straight in a new direction which is not known to the fast boat. How should the pursuing vessel proceed in order to be sure of catching the other boat?The amazing answer is that the pursuing boat should continue to the point where the slow boat would be if it had set its course directly for the pursuing boat when the fog set in. If the boat is not there, it should proceed in a spiral whose origin is the point where the slow boat was when the fog set in. The spiral must be constructed in such a way that, while circling the origin, the fast boat's distance from it increases at the same rate as the boat being pursued. The two courses must therefore intersect before the fast boat has completed one circuit. In order to make the problem reasonably practical, the fast boat should be capable of maintaining a speed four or five times..
The tractrix arises in the following problem posed to Leibniz: What is the path of an object starting off with a vertical offset when it is dragged along by a string of constant length being pulled along a straight horizontal line (Steinhaus 1999, pp. 250-251)? By associating the object with a dog, the string with a leash, and the pull along a horizontal line with the dog's master, the curve has the descriptive name "hundkurve" (dog curve) in German. Leibniz found the curve using the fact that the axis is an asymptote to the tractrix (MacTutor Archive).From its definition, the tractrix is precisely the catenary involute described by a point initially on the vertex (so the catenary is the tractrix evolute). The tractrix is sometimes called the tractory or equitangential curve. The tractrix was first studied by Huygens in 1692, who gave it the name "tractrix." Later, Leibniz, Johann Bernoulli, and others studied the curve.In..
In the mice problem, also called the beetle problem, mice start at the corners of a regular -gon of unit side length, each heading towards its closest neighboring mouse in a counterclockwise direction at constant speed. The mice each trace out a logarithmic spiral, meet in the center of the polygon, and travel a distanceThe first few values for , 3, ..., aregiving the numerical values 0.5, 0.666667, 1, 1.44721, 2, 2.65597, 3.41421, 4.27432, 5.23607, .... The curve formed by connecting the mice at regular intervals of time is an attractive figure called a whirl.The problem is also variously known as the (three, four, etc.) (bug, dog, etc.) problem. It can be generalized to irregular polygons and mice traveling at differing speeds (Bernhart 1959). Miller (1871) considered three mice in general positions with speeds adjusted to keep paths similar and the triangle similar to the original...
The first Brocard point is the interior point (also denoted or ) of a triangle with points labeled in counterclockwise order for which the angles , , and are equal, with the unique such angle denoted . It is not a triangle center, but has trilinear coordinates(1)(Kimberling 1998, p. 47).Note that extreme care is needed when consulting the literature, since reversing the order in which the points of the triangle are labeled results in exchanging the Brocard points.The second Brocard point is the interior point (also denoted or ) for which the angles , , and are equal, with the unique such angle denoted . It is not a triangle center, but has trilinear coordinates(2)(Kimberling 1998, p. 47).Moreover, the two angles are equal, and this angle is called the Brocard angle,(3)(4)The first two Brocard points are isogonal conjugates (Johnson 1929, p. 266). They were described by French army officer Henri Brocard in 1875, although they..
If is a simple closed curve in , then the Jordan curve theorem, also called the Jordan-Brouwer theorem (Spanier 1966) states that has two components (an "inside" and "outside"), with the boundary of each.The Jordan curve theorem is a standard result in algebraic topology with a rich history. A complete proof can be found in Hatcher (2002, p. 169), or in classic texts such as Spanier (1966). Recently, a proof checker was used by a Japanese-Polish team to create a "computer-checked" proof of the theorem (Grabowski 2005).
The polar coordinates (the radial coordinate) and (the angular coordinate, often called the polar angle) are defined in terms of Cartesian coordinates by(1)(2)where is the radial distance from the origin, and is the counterclockwise angle from the x-axis. In terms of and ,(3)(4)(Here, should be interpreted as the two-argument inverse tangent which takes the signs of and into account to determine in which quadrant lies.) It follows immediately that polar coordinates aren't inherently unique; in particular, will be precisely the same polar point as for any integer . What's more, one often allows negative values of under the assumption that is plotted identically to .The expression of a point as an ordered pair is known as polar notation, the equation of a curve expressed in polar coordinates is known as a polar equation, and a plot of a curve in polar coordinates is known as a polar plot.In much the same way that Cartesian curves can be plotted on..
A plot of a function expressed in spherical coordinates, with radius as a function of angles and . Polar plots can be drawn using SphericalPlot3D[r, phi, phimin, phimax, theta, thetamin, thetamax]. The plots above are spherical plots of the equations and , where denotes the real part and the imaginary part. The spherical plot of a constant is a sphere of radius .
A plot of a function expressed in polar coordinates, with radius as a function of angle . Polar plots can be drawn in the Wolfram Language using PolarPlot[r, t, tmin, tmax]. The plot above is a polar plot of the polar equation , giving a cardioid.Polar plots of give curves known as roses, while polar plots of produce what's known as Archimedes' spiral, a special case of the Archimedean spiral corresponding to . Other specially-named Archimedean spirals include the lituus when , the hyperbolic spiral when , and Fermat's spiral when . Note that lines and circles are easily-expressed in polar coordinates as(1)and(2)for the circle with center and radius , respectively. Note that equation () is merely a particular instance of the equation(3)defining a conic section of eccentricity and semilatus rectum . In particular, the circle is the conic of eccentricity , while yields a general ellipse, a parabola, and a hyperbola.The plotting of a complex number..
The polar curve(1)that can be used for angle trisection. It was devised by Ceva in 1699, who termed it the cycloidum anomalarum (Loomis 1968, p. 29). It has Cartesian equation(2)It has area(3)and arc length(4)(5)(OEIS A138497), with , where , , and are complete elliptic integrals of the first, second, and third, respectively.The arc length function is a slightly complicated expression that can be expressed in closed form in terms of elliptic functions, and the curvature is given by(6)
The 60 Pascal lines of a hexagon inscribed in a conic intersect three at a time through 20 Steiner points, and also three at a time in 60 points known as Kirkman points. Each Steiner point lies together with three Kirkman points on a total of 20 lines known as Cayley lines. There is a reciprocity relationship between the 60 Kirkman points and the 60 Pascal lines (Hesse, quoted in Salmon 1960), although the relationship is not one of duality in the commonly accepted meaning of that word.
If two single-valued continuous functions (curvature) and (torsion) are given for , then there exists exactly one space curve, determined except for orientation and position in space (i.e., up to a Euclidean motion), where is the arc length, is the curvature, and is the torsion.
The golden angle is the angle that divides a full angle in a golden ratio (but measured in the opposite direction so that it measures less than ), i.e.,(1)(2)(3)(4)(5)(6)(7)(OEIS A131988 and A096627;Livio 2002, p. 112).It is implemented in the Wolfram Languageas GoldenAngle.van Iterson showed in 1907 that points separated by on a tightly bound spiral tends to produce interlocked spirals winding in opposite directions, and that the number of spirals in these two families tend to be consecutive Fibonacci numbers (Livio 2002, p. 112).Another angle related to the golden ratio is theangle(8)or twice this angle(9)the later of which is the smaller interior angle in the goldenrhombus.
The Ochoa curve is the elliptic curvegiven in Weierstrass form asThe complete set of 23 integer solutions (where solutions of the form are counted as a single solution) to this equation consists of , (, 4520), (, 13356), (, 14616), (, 10656), (91, 8172), (227, 4228), (247, 3528), (271, 2592), (455, 200), (499, 3276), (523, 4356), (530, 4660), (599, 7576), (751, 14112), (1003, 25956), (1862, 75778), (3511, 204552), (5287, 381528), (23527, 3607272), (64507, 16382772), (100102, 31670478), and (1657891, 2134685628) (OEIS A141144 and A141145; Stroeker and de Weger 1994).
An elliptic curve of the form for an integer. This equation has a finite number of solutions in integers for all nonzero . If is a solution, it therefore follows that is as well.Uspensky and Heaslet (1939) give elementary solutions for , , and 2, and then give , , , and 1 as exercises. Euler found that the only integer solutions to the particular case (a special case of Catalan's conjecture) are , , and . This can be proved using Skolem's method, using the Thue equation , using 2-descent to show that the elliptic curve has rank 0, and so on. It is given as exercise 6b in Uspensky and Heaslet (1939, p. 413), and proofs published by Wakulicz (1957), Mordell (1969, p. 126), Sierpiński and Schinzel (1988, pp. 75-80), and Metsaenkylae (2003).Solutions of the Mordell curve with are summarized in the table below for small .solutions123456none7none8910Values of such that the Mordell curve has no integer solutions are given by 6, 7, 11, 13,..
There are two incompatible definitions of the squircle.The first defines the squircle as the quartic plane curve which is special case of the superellipse with and , namely(1)illustrated above. This curve as arc length(2)(3)(OEIS A186642), where is a Meijer G-function (M. Trott, pers. comm., Oct. 21, 2011), encloses area(4)and has area moment of inertia tensor(5)The second definition of the squircle was given by Fernandez Guasti (1992), but apparently not dubbed with the name "squircle" until later (Fernández Guasti et al. 2005). This curve has quartic Cartesian equation(6)with squareness parameter , where corresponds to a circle with radius and to a square of side length . This curve is actually semialgebraic, as it must be restricted to to exclude other branches. This squircle encloses area(7)where is an elliptic integral of the second kind, which can be verified reduces to for and for .Both versions..
By analogy with the squircle, a term first apparently used by Fernández Guasti et al. (2005), the term "rectellipse" (used here for the first time) is a natural generalization to the case of unequal vertical and horizontal dimensions.The first definition of the rectellipse is the quartic plane curve which is special case of the superellipse with , namely(1)illustrated above. This curve encloses area(2)and has area moment of inertia tensor(3)The second definition of the rectellipse was given, though not explicitly named, by Fernandez Guasti (1992). This curve has quartic Cartesian equation(4)with squareness parameter , where corresponds to an ellipse with semiaxes and and to a rectangle the side lengths and . This curve is actually semialgebraic, as it must be restricted to and to exclude other branches. This rectellipse encloses area(5)where is an elliptic integral of the second kind, which can be verified reduces..
The only known classically known algebraic curve of curve genus that has an explicit parametrization in terms of standard special functions (Burnside 1893, Brezhnev 2001). This equation is given by(1)The closed portion of the curve has area(2)(3)where is a gamma function.The closed portion of this curve has a parametrization in terms of the Weierstrasselliptic function given by(4)(5)where(6)the half-periods are given by and ranges over complex values (Brezhnev 2001).
Algebraic geometry is the study of geometries that come from algebra, in particular, from rings. In classical algebraic geometry, the algebra is the ring of polynomials, and the geometry is the set of zeros of polynomials, called an algebraic variety. For instance, the unit circle is the set of zeros of and is an algebraic variety, as are all of the conic sections.In the twentieth century, it was discovered that the basic ideas of classical algebraic geometry can be applied to any commutative ring with a unit, such as the integers. The geometry of such a ring is determined by its algebraic structure, in particular its prime ideals. Grothendieck defined schemes as the basic geometric objects, which have the same relationship to the geometry of a ring as a manifold to a coordinate chart. The language of category theory evolved at around the same time, largely in response to the needs of the increasing abstraction in algebraic geometry.As a consequence,..
An algebraic curve over a field is an equation , where is a polynomial in and with coefficients in . A nonsingular algebraic curve is an algebraic curve over which has no singular points over . A point on an algebraic curve is simply a solution of the equation of the curve. A -rational point is a point on the curve, where and are in the field .The following table lists the names of algebraic curves of a given degree.ordercurveexamples2quadratic curvecircle, ellipse, hyperbola, parabola3cubic curvecissoid of Diocles, conchoid of de Sluze, folium of Descartes, Maclaurin trisectrix, Maltese cross curve, Mordell curve, Ochoa curve, right strophoid, semicubical parabola, serpentine curve, Tschirnhausen cubic, witch of Agnesi4quartic curveampersand curve, bean curve, bicorn, bicuspid curve, bifoliate, bifolium, bitangent-rich curve, bow, bullet nose, butterfly curve, capricornoid, cardioid, Cartesian ovals, Cassini ovals, conchoid..