A syntractrix is a curve of the form
- [1]
It is the locus of a point on the tangent of a tractrix at a constant distance from the point of tangency, as the point of tangency is moved along the curve. [2]
A syntractrix is a curve of the form
It is the locus of a point on the tangent of a tractrix at a constant distance from the point of tangency, as the point of tangency is moved along the curve. [2]
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity.
In physics and geometry, a catenary is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends.
In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.
In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena, through Fourier analysis.
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c if the line passes through the point (c, f ) on the curve and has slope f'(c), where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions defined for the hyperbola rather than on the circle: just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the equilateral hyperbola.
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
A helix, plural helixes or helices, is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, and many proteins have helical substructures, known as alpha helices. The word helix comes from the Greek word ἕλιξ, "twisted, curved". A "filled-in" helix – for example, a "spiral" (helical) ramp – is called a helicoid.
A hyperbolic spiral is a plane curve, which can be described in polar coordinates by the equation
In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.
In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.
In geometry, the subtangent and related terms are certain line segments defined using the line tangent to a curve at a given point and the coordinate axes. The terms are somewhat archaic today but were in common use until the early part of the 20th century
A tractrix is the curve along which an object moves, under the influence of friction, when pulled on a horizontal plane by a line segment attached to a tractor (pulling) point that moves at a right angle to the initial line between the object and the puller at an infinitesimal speed. It is therefore a curve of pursuit. It was first introduced by Claude Perrault in 1670, and later studied by Isaac Newton (1676) and Christiaan Huygens (1692).
A Cassini oval is a quartic plane curve defined as the set of points in the plane such that the product of the distances to two fixed points is constant. This may be contrasted with an ellipse, for which the sum of the distances is constant, rather than the product. Cassini ovals are the special case of polynomial lemniscates when the polynomial used has degree 2.
In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve defined by an equation of the form
The smoothed octagon is a region in the plane conjectured to have the lowest maximum packing density of the plane of all centrally symmetric convex shapes. It is constructed by replacing the corners of a regular octagon with a section of a hyperbola that is tangent to the two sides adjacent to the corner and asymptotic to the sides adjacent to these.
In integral calculus, the Weierstrass substitution or tangent half-angle substitution is a method for evaluating integrals, which converts a rational function of trigonometric functions of into an ordinary rational function of by setting . No generality is lost by taking these to be rational functions of the sine and cosine. The general transformation formula is
In the geometry of curves, an orthoptic is the set of points for which two tangents of a given curve meet at a right angle.
In mathematics, a conical spiral is a curve on a right circular cone, whose floor plan is a plane spiral. If the floor plan is a logarithmic spiral, it is called conchospiral.
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