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In it, geometrical shapes can be made, as well as expressions from the normal graphing calculator, with extra features. [8] In September 2023, Desmos released a beta for a 3D calculator, which added features on top of the 2D calculator, including cross products, partial derivatives and double-variable parametric equations. [9]
Parametric equations are commonly used in kinematics, where the trajectory of an object is represented by equations depending on time as the parameter. Because of this application, a single parameter is often labeled t ; however, parameters can represent other physical quantities (such as geometric variables) or can be selected arbitrarily for ...
Top: Output signal as a function of time. Middle: Input signal as a function of time. Bottom: Resulting Lissajous curve when output is plotted as a function of the input. In this particular example, because the output is 90 degrees out of phase from the input, the Lissajous curve is a circle, and is rotating counterclockwise.
The curve is given by the following parametric equations: [2] ... In 2006, two mathematicians using Mathematica analyzed the function, ...
It is the locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity. Equivalently, in polar coordinates ( r , θ ) it can be described by the equation r = b ⋅ θ {\displaystyle r=b\cdot \theta } with real number b .
The mathematical basis for Bézier curves—the Bernstein polynomials—was established in 1912, but the polynomials were not applied to graphics until some 50 years later when mathematician Paul de Casteljau in 1959 developed de Casteljau's algorithm, a numerically stable method for evaluating the curves, and became the first to apply them to computer-aided design at French automaker Citroën ...
The equation defining a plane curve expressed in polar coordinates is known as a polar equation. In many cases, such an equation can simply be specified by defining r as a function of φ. The resulting curve then consists of points of the form (r(φ), φ) and can be regarded as the graph of the polar function r.
Examples of superellipses for =, =. A superellipse, also known as a Lamé curve after Gabriel Lamé, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but defined by an equation that allows for various shapes between a rectangle and an ellipse.