Parametric equation in the context of "Parametrization (geometry)"

A parametric surface is a surface in the Euclidean space ${\displaystyle \mathbb {R} ^{3}}$ which is defined by a parametric equation with two parameters ⁠${\displaystyle \mathbf {r} :\mathbb {R} ^{2}\to \mathbb {R} ^{3}}$⁠. Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem, and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization.

In geometry, an epitrochoid (/ɛpɪˈtrɒkɔɪd/ or /ɛpɪˈtroʊkɔɪd/) is a roulette traced by a point attached to a circle of radius r rolling around the outside of a fixed circle of radius R, where the point is at a distance d from the center of the exterior circle.

The parametric equations for an epitrochoid are:

In theoretical chemistry, an energy profile is a theoretical representation of a chemical reaction or process as a single energetic pathway as the reactants are transformed into products. This pathway runs along the reaction coordinate, which is a parametric curve that follows the pathway of the reaction and indicates its progress; thus, energy profiles are also called reaction coordinate diagrams. They are derived from the corresponding potential energy surface (PES), which is used in computational chemistry to model chemical reactions by relating the energy of a molecule(s) to its structure (within the Born–Oppenheimer approximation).

Qualitatively, the reaction coordinate diagrams (one-dimensional energy surfaces) have numerous applications. Chemists use reaction coordinate diagrams as both an analytical and pedagogical aid for rationalizing and illustrating kinetic and thermodynamic events. The purpose of energy profiles and surfaces is to provide a qualitative representation of how potential energy varies with molecular motion for a given reaction or process.

In mathematics, the Tschirnhausen cubic is a cubic plane curve defined by the polar equation$${\displaystyle r=a\sec ^{3}\left({\frac {\theta }{3}}\right)}$$or the equivalent algebraic equation$${\displaystyle 27ay^{2}=(a-x)(8a+x)^{2}.}$$

It is a nodal cubic, meaning that it crosses itself at one point, its node. The angle at this crossing point, inside the loop formed by the crossing, is 60°. Because the Tschirnhausen cubic has this singularity, it can be given a parametric equation, and any arc of it can be drawn as a cubic Bézier curve. It is a special case of a sinusoidal spiral, of a pursuit curve, and of a Pythagorean hodograph curve.

Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus.

Many specific curves have been thoroughly investigated using the synthetic approach. Differential geometry takes another approach: curves are represented in a parametrized form, and their geometric properties and various quantities associated with them, such as the curvature and the arc length, are expressed via derivatives and integrals using vector calculus. One of the most important tools used to analyze a curve is the Frenet frame, a moving frame that provides a coordinate system at each point of the curve that is "best adapted" to the curve near that point.

In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of singular point of a curve.

For a plane curve defined by an analytic, parametric equation

Nonparametric regression is a form of regression analysis where the predictor does not take a predetermined form but is completely constructed using information derived from the data. That is, no parametric equation is assumed for the relationship between predictors and dependent variable. A larger sample size is needed to build a nonparametric model having the same level of uncertainty as a parametric model because the data must supply both the model structure and the parameter estimates.