Glossary
Direction Vector
A vector (x2 - x1, y2 - y1) that indicates the magnitude and direction of movement along a line segment from a starting point (x1, y1) to an ending point (x2, y2).
Example:
For a car traveling from (1,2) to (5,8), the direction vector would be (5-1, 8-2) = (4,6), showing its displacement.
Domain (of Parameter for Unit Circle)
The specific range of values for the parameter 't' (typically 0 ≤ t ≤ 2π) that completes one full revolution around the unit circle.
Example:
To ensure a full circle is drawn, the domain for 't' in x(t)=cos(t), y(t)=sin(t) must be 0 to 2π.
General Parametric Equation for a Circle
The equations x(t) = a + rcos(t) and y(t) = b + rsin(t), which describe a circle with center (a, b) and radius r.
Example:
To model the path of a Ferris wheel with a radius of 10 meters and its center 12 meters above the ground, you would use the general parametric equation for a circle with r=10 and (a,b)=(0,12).
Parameter
The independent variable (often 't' or 'k') used in parametric equations to define both the x and y coordinates, effectively tracing a path over time or a specific interval.
Example:
In the equations x(t) = 3t and y(t) = t^2, 't' is the parameter that determines the position (x,y) of a moving object at any given moment.
Parametric Equations
A method to describe motion or paths where both x and y coordinates are expressed in terms of a third independent variable, often 't' (the parameter).
Example:
To model the path of a projectile, you might use parametric equations like x(t) = (initial velocity * cos(angle)) * t and y(t) = (initial velocity * sin(angle)) * t - 0.5 * g * t^2.
Parametric Equations for a Line Segment
Equations of the form x = x1 + k(x2 - x1) and y = y1 + k(y2 - y1), where (x1, y1) is the start point, (x2, y2) is the end point, and the parameter 'k' typically ranges from 0 to 1.
Example:
To plot a straight path for a drone from (0,0) to (10,5), you would use the parametric equations for a line segment with (x1,y1)=(0,0) and (x2,y2)=(10,5), and 0 ≤ k ≤ 1.
Parametrically Defined Circle
A circle whose points (x, y) are described by equations where both x and y are functions of a parameter, typically 't', showing movement around the circle.
Example:
The path of a satellite orbiting Earth in a perfect circle can be represented as a parametrically defined circle, showing its position (x, y) at any given time 't'.
Parametrically Defined Line
A line or line segment whose points (x, y) are described by equations where both x and y are functions of a parameter, typically 't' or 'k', showing movement along the line.
Example:
A robot moving in a straight line from point A to point B can have its path described as a parametrically defined line segment, showing its exact location at each moment.
Rotation (in Parametric Equations)
Modifying the parameter 't' by adding a constant (e.g., t + c) within the trigonometric functions to shift the starting point or orientation of a parametrically defined circle.
Example:
If a satellite's orbit needs to start at a different angle, you can apply a rotation by adjusting the 't' value in its parametric equations, like x(t) = cos(t + π/4).
Standard Parametric Equations (for Unit Circle)
The specific parametric equations x(t) = cos(t) and y(t) = sin(t) that describe a point moving counterclockwise around the unit circle.
Example:
If you want to trace a simple circle on a graph, you'd use the standard parametric equations (x(t) = cos(t), y(t) = sin(t)) with t from 0 to 2π.
Unit Circle
A circle with a radius of 1 unit, centered at the origin (0,0) in the Cartesian coordinate system, fundamental for understanding trigonometric values.
Example:
When learning trigonometry, the unit circle is fundamental for understanding sine and cosine values for various angles.