### Key Terms

- angular coordinate
- $\theta $ the angle formed by a line segment connecting the origin to a point in the polar coordinate system with the positive radial (
*x*) axis, measured counterclockwise

- cardioid
- a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius; the equation of a cardioid is $r=a\left(1+\text{sin}\phantom{\rule{0.2em}{0ex}}\theta \right)$ or $r=a\left(1+\text{cos}\phantom{\rule{0.2em}{0ex}}\theta \right)$

- conic section
- a conic section is any curve formed by the intersection of a plane with a cone of two nappes

- cusp
- a pointed end or part where two curves meet

- cycloid
- the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage

- directrix
- a directrix (plural: directrices) is a line used to construct and define a conic section; a parabola has one directrix; ellipses and hyperbolas have two

- discriminant
- the value $4AC-{B}^{2},$ which is used to identify a conic when the equation contains a term involving $xy,$ is called a discriminant

- eccentricity
- the eccentricity is defined as the distance from any point on the conic section to its focus divided by the perpendicular distance from that point to the nearest directrix

- focal parameter
- the focal parameter is the distance from a focus of a conic section to the nearest directrix

- focus
- a focus (plural: foci) is a point used to construct and define a conic section; a parabola has one focus; an ellipse and a hyperbola have two

- general form
- an equation of a conic section written as a general second-degree equation

- limaçon
- the graph of the equation $r=a+b\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta $ or $r=a+b\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta .$ If $a=b$ then the graph is a cardioid

- major axis
- the major axis of a conic section passes through the vertex in the case of a parabola or through the two vertices in the case of an ellipse or hyperbola; it is also an axis of symmetry of the conic; also called the transverse axis

- minor axis
- the minor axis is perpendicular to the major axis and intersects the major axis at the center of the conic, or at the vertex in the case of the parabola; also called the conjugate axis

- nappe
- a nappe is one half of a double cone

- orientation
- the direction that a point moves on a graph as the parameter increases

- parameter
- an independent variable that both
*x*and*y*depend on in a parametric curve; usually represented by the variable*t*

- parameterization of a curve
- rewriting the equation of a curve defined by a function $y=f\left(x\right)$ as parametric equations

- parametric curve
- the graph of the parametric equations $x\left(t\right)$ and $y\left(t\right)$ over an interval $a\le t\le b$ combined with the equations

- parametric equations
- the equations $x=x\left(t\right)$ and $y=y\left(t\right)$ that define a parametric curve

- polar axis
- the horizontal axis in the polar coordinate system corresponding to $r\ge 0$

- polar coordinate system
- a system for locating points in the plane. The coordinates are $r,$ the radial coordinate, and $\theta ,$ the angular coordinate

- polar equation
- an equation or function relating the radial coordinate to the angular coordinate in the polar coordinate system

- pole
- the central point of the polar coordinate system, equivalent to the origin of a Cartesian system

- radial coordinate
- $r$ the coordinate in the polar coordinate system that measures the distance from a point in the plane to the pole

- rose
- graph of the polar equation $r=a\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}2\theta $ or $r=a\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}2\theta $ for a positive constant
*a*

- space-filling curve
- a curve that completely occupies a two-dimensional subset of the real plane

- standard form
- an equation of a conic section showing its properties, such as location of the vertex or lengths of major and minor axes

- vertex
- a vertex is an extreme point on a conic section; a parabola has one vertex at its turning point. An ellipse has two vertices, one at each end of the major axis; a hyperbola has two vertices, one at the turning point of each branch