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?

theta for the angle, numbers, brackets, + - * / ^, and the functions and constants listed under the plot: sin, cos, tan, sec, csc, cot, asin, acos, atan, sinh, cosh, tanh, sqrt, abs, exp, ln, log, floor, ceil, round, sign, min and max, with pi, tau and e as constants. Multiplication can be left out where it is obvious, so 2theta, 3 cos(theta) and sin(3 theta) all read the way you would write them on paper. A name outside that list, an unclosed bracket or a stray character each produce their own message saying what to change.

Can I work in degrees instead of radians?

Yes, and it is a real mode rather than a relabelling. In degrees a full turn is 0 to 360, and sin, cos and tan take their argument in degrees too, so sin(theta) peaks at theta 90 exactly as it peaks at pi/2 in radians. The range fields accept expressions in either mode, so pi, 2pi and 3pi/2 are all valid entries in radians, and the readout prints angles back as multiples of pi where they are one.

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?

Copy SVG puts the drawing on the clipboard as markup and Download SVG saves it as a file, both built from the same scene the stage draws, so the file cannot disagree with the picture. Each arc of the curve is its own path in that file, which is how the gaps survive being pasted somewhere else. Copy samples as CSV gives you the coordinate table as theta, r, x and y, with any angle where r has no value written out as such rather than left blank.

What will this page not do?

It plots r against theta and reports what it sampled. It does not rearrange an equation for you, work out where two curves meet, or turn the drawing into a proof: the shape name is a match against a list of standard forms and nothing more. It also draws one expression at a time, so comparing two curves means plotting them one after the other.