Chapter 1: Parametric Equations and Polar Coordinates

angular coordinate

𝜃 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

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

(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 4⁢𝐴⁢𝐶−𝐵2, which is used to identify a conic when the equation contains a term involving 𝑥⁢𝑦,

eccentricity

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 distance from a focus of a conic section to the nearest directrix

focus

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⁢sin⁡𝜃 or 𝑟=𝑎+𝑏⁢cos⁡𝜃. If 𝑎=𝑏 then the graph is a cardioid

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

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

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⁡(𝑥) as parametric equations

parametric curve

the graph of the parametric equations 𝑥⁡(𝑡) and 𝑦⁡(𝑡) over an interval 𝑎≤𝑡≤𝑏 combined with the equations

parametric equations

the equations 𝑥=𝑥⁡(𝑡) and 𝑦=𝑦⁡(𝑡) that define a parametric curve

polar axis

the horizontal axis in the polar coordinate system corresponding to 𝑟≥0

polar coordinate system

a system for locating points in the plane. The coordinates are 𝑟, the radial coordinate, and 𝜃, 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⁢cos⁡2⁢𝜃 or 𝑟=𝑎⁢sin⁡2⁢𝜃 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

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

Derivative of parametric equations

Second-order derivative of parametric equations

Area under a parametric curve

Arc length of a parametric curve

Surface area generated by a parametric curve

Area of a region bounded by a polar curve

Arc length of a polar curve