01 Learning Polar Graphing
- Author:
- Auston B Cron
A polar grid consists of a sequence of concentric circles that are divided radially by various radial lines at regular angular divisions usually in increments of 5°, 10°, or 15°. Periodically you may want Press the Recycle Icon on the upper right to clear the worksheet.
GeoGebra contains three commands that will assist us in creating the polar grid.
Circle[Point M, Number r], Line[Point, Direction vector v], and
Sequence[Expression, Variable i, Number a, Number b, <Increment>]
To draw a circle with a center at Point(4,5) and radius = 3, we would enter the following command in the input line: Circle[(4, 5), 3] {You should try this entry and others prior to proceeding.}
To draw a line that starts at the point (4, 5) toward the end of the direction vector of 4x + 5y = 12 or (5, -4), we type the following Line[(4, 5),(5, -4)] in the input line. "A line with equation ax + by = c has the Direction vector (b, -a)." {You should try this entry and others prior to proceeding.}
To draw a line through the origin at a 15° angle type: Line[(0, 0), (cos(15°), sin(15°))]
The final important command is Sequence. This command allows incremental copies of an expression to be graphed. To create a simple family of parabolas type: Sequence[a x2, a, -2, 2]. The sequence automatically increments by ones, this can be change by typing: Sequence[a x2, a, -2, 2, 0.5]
The following command will create 40 concentric circle centered at the origin by 1/2 unit increments:
polarCircles = Sequence[Circle[(0, 0), k / 2], k, 1, 20]
The following command will create a series of radial lines at increments from 0 through 175° in 15° increments:
radialLines = Sequence[Line[(0, 0), (cos(a), sin(a))], a, 0, 175°, 15°]
The Parametric Curve Command allows us to create or graph polar functions. Curve[Expression e1, Expression e2, Parameter t, Number a, Number b]: Yields the Cartesian parametric curve for the given x-expression e1 and y-expression e2 (using parameter t) within the given interval [a, b].
r(x) = 1 {hide this line, this function will allow you to enter polar equations}
Curve[r(i) cos(i), r(i) sin(i), i, 0, 2 pi] {this will draw a circle of radius one}