Polar Graphing Calculator

Common questions

How do I plot a cardioid?

Type 1 + cos(theta) and leave the range at 0 to 2pi. The page names it a cardioid, reports r max 2.00 at theta 0 and r min 0.00 at theta pi, and draws it as one unbroken arc. The name is a heuristic match: the parsed expression is compared against a short list of standard polar forms, so 1 + cos(theta) and 1 - 2 sin(theta) are named a cardioid and a limaçon with an inner loop, while an expression outside that list is drawn and left unnamed rather than guessed at.

Why does my curve have gaps in it?

Because r left the plot or had no value there, and joining the two sides would draw a line that is not part of the curve. tan(theta) is the clear case: r runs away near a quarter turn and a full three quarters of a turn, so the page cuts the arc at the edge of the plot and starts a new one on the other side, three arcs with two gaps. Where r stays in a sensible range the plot reaches the largest sample and nothing is cut; where a few samples run away, the edge is set from the bulk of them instead, so a handful of enormous values near an asymptote cannot flatten the rest of the curve into a dot at the middle.

How many petals does a rose curve have?

It depends on whether the multiplier on theta is odd or even. sin(3 theta) is named a three-petal rose here, and it is complete in half a turn; sin(2 theta) is a four-petal rose and needs the full turn. That is the odd and even rule: k petals for odd k, 2k petals for even k. If the range you set is too short for the whole shape, the name says so, for example part of a four-petal rose from 0 to pi.

What can I type in the expression?

Enter the right-hand side of r = f(theta), without r =. Use theta, t or θ for the angle; numbers, brackets, + - * / ^; and the listed functions and constants. Examples include sin(3 theta), 2cos(theta) and theta/4. Multiplication can be omitted in 2theta and 3 cos(theta). Functions need brackets, such as sin(theta). pi, tau and e are constants. An unknown name, missing bracket or unsupported character produces a message explaining what to change. The page does not solve implicit equations for r.

Can I work in degrees instead of radians?

Yes. Choose Degrees and use 0 to 360 for one full turn, or Radians and use 0 to 2pi. After changing the unit, the 1 turn button resets both range fields for that unit. Trigonometric functions in the expression and range fields use the selected unit; inverse trigonometric functions return angles in that unit. For example, asin(1) is 90 in degrees and pi/2 in radians. pi itself remains the number 3.14159... in either mode.

How many samples does it take, and does that change the shape?

720 steps across the range by default, which is 721 evenly spaced angles counting both ends, and you can set anything from 32 to 4,000 steps. Those even samples are what r max, r min and the coordinate table report. On top of them the page works out extra points wherever two neighbours land far apart on the page, which is why a tight curve like sin(10 theta) ends up with more plotted points than samples. A feature narrower than the gap between two samples can still be missed, so raise the count when you want to look closely at something small.

Can I keep the graph?

Download SVG hands the graph to your browser as an SVG file. Copy SVG copies its markup. Both use the same sampled paths as the displayed curve, with one path per arc so gaps are preserved. Copy samples as CSV copies the 25-row coordinate table as theta, r, x and y; undefined values are written as no value. The page does not export PDF or PNG.

What will this page not do?

It graphs one explicit r(theta) expression at a time on a two-dimensional polar grid. It does not accept a list of arbitrary polar points, plot 3D surfaces, solve an implicit equation for r, differentiate, or work out where two curves meet. Standard shape names are heuristic matches, and r max and r min are sampled readings rather than solved extrema.

How are polar coordinates converted to x and y?

At each sampled angle, the page evaluates r and uses x = r cos(theta) and y = r sin(theta). theta is measured from the positive x-axis, counterclockwise for positive angles. A negative r puts the point in the opposite direction. In degree mode, the angle is converted to radians for these Cartesian formulas. The table shows points sampled from your equation, rather than points you enter individually.

Is this an online polar graphing calculator I can use without an account?

Yes. The core graphing controls are free to use on this page without an account or installation. Your expression is calculated in the browser. This page is an independent calculator and does not open or control Desmos or Wolfram.