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Definition of slope angle and sector Animation showing the constant angle between an intersecting circle centred at the origin and a logarithmic spiral. The logarithmic spiral r = a e k φ , k ≠ 0 , {\displaystyle r=ae^{k\varphi }\;,\;k\neq 0,} has the following properties (see Spiral ):
The Fermat spiral with polar equation = can be converted to the Cartesian coordinates (x, y) by using the standard conversion formulas x = r cos φ and y = r sin φ.Using the polar equation for the spiral to eliminate r from these conversions produces parametric equations for one branch of the curve:
Equivalently, in polar coordinates (r, θ) it can be described by the equation = with real number b. Changing the parameter b controls the distance between loops. From the above equation, it can thus be stated: position of the particle from point of start is proportional to angle θ as time elapses.
The equation of the circle determined by three points (,), (,), (,) not on a line is obtained by a conversion of the 3-point form of a circle equation: () + () () () = () + () () (). Homogeneous form In homogeneous coordinates , each conic section with the equation of a circle has the form x 2 + y 2 − 2 a x z − 2 b y z + c z 2 = 0 ...
The trifolium curve (also three-leafed clover curve, 3-petaled rose curve, and paquerette de mélibée) is a type of quartic plane curve. The name comes from the Latin terms for 3-leaved, defining itself as a folium shape with 3 equally sized leaves. It is described as + + + = By solving for y, the curve can be described by the following function:
For a hyperbolic spiral with equation =, a circular arc centered at the origin, continuing clockwise for length from any of its points, will end on the -axis. [3] Because of this equal-length property, the starting marks of 200m and 400m footraces are placed in staggered positions along a hyperbolic spiral.
Cissoid of Diocles traced by points M with ¯ = ¯ Animation visualizing the Cissoid of Diocles. In geometry, the cissoid of Diocles (from Ancient Greek κισσοειδής (kissoeidēs) 'ivy-shaped'; named for Diocles) is a cubic plane curve notable for the property that it can be used to construct two mean proportionals to a given ratio.
Construction of the limaçon r = 2 + cos(π – θ) with polar coordinates' origin at (x, y) = ( 1 / 2 , 0). In geometry, a limaçon or limacon / ˈ l ɪ m ə s ɒ n /, also known as a limaçon of Pascal or Pascal's Snail, is defined as a roulette curve formed by the path of a point fixed to a circle when that circle rolls around the outside of a circle of equal radius.