How an Equation Places a Polar Point

Direction, then distance

Start at the bright point on the right. The origin is where the axes meet. This point is four units from the origin, in the direction we call angle zero. A polar coordinate gives these two instructions: choose a direction, then travel a signed distance along it. The faint outline shows the complete curve we will trace.

Now the direction turns upward. Watch the marked point move as the angle approaches a quarter turn. The distance changes too: it falls from four to two. We are not moving around a circle of fixed radius. The equation supplies a new distance for every direction.

At a quarter turn the point is two units above the origin. The Greek letter theta names the angle. Here theta is pi divided by two radians, which is ninety degrees. You can read direction from the ray and distance from the marked segment; neither coordinate is the point’s ordinary horizontal position.

Follow the direction back toward the right. The point moves outward as it returns. Angle and radius can change together, but they do different jobs. Keep that distinction as we bring the rule into focus: the angle tells us where to aim, and the signed radius tells us where to stop.

A rule supplies the radius

The displayed rule is r equals two plus two cosine theta. You do not need to have used cosine before. For this example, we can begin with a few of its values. At angle zero, cosine is one. Two plus two times one gives radius four, exactly where the point sits.

At a quarter turn, cosine is zero. Two plus two times zero leaves radius two. Watch the same point arrive above the origin again. The formula describes the distance you have just seen; it is not a separate picture that we must memorize without a referent.

Continue to half a turn. Cosine now reaches minus one, so two plus two times minus one is zero. The bright point arrives at the origin. A direction still exists, but traveling zero distance leaves us at the origin, whatever direction we chose.

Look closely at that meeting point. The curve has a pointed cusp here. Its radius has reached zero without becoming negative: the smallest value of two plus two cosine theta is zero. The cusp is a visible consequence of that minimum. We will soon compare it with a rule whose minimum goes below zero.

Complete the journey

Move past the cusp toward three quarters of a turn. The radius grows again, reaching two when the direction points down. The marker is now below the origin. The same radius of two occurred above the origin, but the different direction places this point somewhere else.

At this lower point, compare the ray with its earlier upward direction. A radius by itself does not locate a point. Two units could put us anywhere on a circle around the origin; the angle selects one of those directions. Here it selects the downward one.

Finish the turn. The direction returns to the right and the radius returns to four. The bright trace now closes around the whole cardioid. We have followed one continuous rule through the right endpoint, the upper point, the cusp, the lower point, and back to the beginning.

Trace the circuit once more, this time watching how direction and distance cooperate. The small rectangular graph follows the same changing radius against angle. It is another view of this rule. Our next lesson will connect the two graphs directly; for now, keep your attention on how the marked polar point is placed.

A signed distance can reverse direction

Change the constant from two to one. The new rule is r equals one plus two cosine theta. At angle zero the radius is three, so the starting point is three units to the right. The new outline has a small inner loop. To explain that loop, we need the sign of the radius.

Turn toward one hundred twenty degrees. Cosine reaches minus one half here, making one plus two times minus one half equal zero. The moving point reaches the origin. This is the first zero of the radius on this circuit; beyond it, the rule begins producing negative values.

Continue to half a turn. The direction ray points left, but radius minus one sends the marked point one unit in the opposite direction, to the right. A negative radius is an instruction to reverse along the line. It does not describe a negative physical length.

Keep going until the direction reaches two hundred forty degrees. The radius returns to zero and the point comes back to the origin. Between these two origin visits, the negative radii traced the inner loop. The below-axis interval in the small graph records the same sign change.

Radius is not the horizontal coordinate

Use a constant radius of two and begin on the right. The point is two units horizontally from the origin and zero units vertically. At this particular angle, the horizontal coordinate equals the radius. That equality comes from aiming along the horizontal axis; it is not the meaning of radius at every angle.

Turn to forty-five degrees while keeping the radius two. The point moves diagonally upward. Its horizontal and vertical coordinates are now equal, each square root of two, approximately one point four one. Neither coordinate is two. The diagonal segment from the origin still has length two, as required by the unchanged polar radius.

Hold this diagonal position. The two perpendicular coordinate distances form the legs of a right triangle whose hypotenuse is the radius. Squaring and adding them gives two plus two, or four, which equals the radius squared. This arithmetic is a check on the point shown, not a claim that the World has added a separate triangle drawing.

Continue to a quarter turn. The horizontal coordinate falls to zero and the vertical coordinate becomes two. Radius has stayed two throughout. The horizontal and vertical amounts are components of the same displacement, so changing direction redistributes those components without changing the length of the segment from the origin.

Turn through the left half of the circle to three quarters of a turn. Above, left and below all use the same positive radius. A point appearing below the horizontal axis does not by itself indicate a negative radius. Here the downward direction and a positive distance put it below the origin.

Change the signed distance without turning

Aim upward again with radius two. We will hold the angle at a quarter turn and change only the constant radius. The rule is constant in angle because the cosine coefficient is zero. Changing its constant changes the complete circular outline as well as the selected point, so watch the ray and marker separately.

Decrease the constant from two to zero. The point travels straight down its fixed upward ray toward the origin. The circle shrinks at the same time. No angular turn caused this motion: every intermediate position uses the same theta, with a smaller nonnegative value of the signed radius.

Continue the constant from zero to minus two. The point passes through the origin and moves below it, opposite to the upward direction. The negative sign reverses the ray instruction. Its physical distance from the origin grows to two even though the signed radius becomes minus two.

Keep the downward point in view. The absolute value of the radius is its distance from the origin: the absolute value of minus two is two. The sign retains extra information about the direction instruction. Discarding that sign before placing the point would incorrectly put this example two units above the origin.

Compare positive radius two at three quarters of a turn. The point stays in that same lower location, but now the direction itself points down and the radius is positive. These two coordinate descriptions differ by a half-turn and a sign reversal. They describe one location through two consistent sets of polar instructions.

One location, different instructions

Return to the middle of the inner loop. At theta equal to pi, radius minus one places the point at one unit to the right. Notice both pieces of information: the direction points left and the negative sign reverses it. Looking only at which side of the screen the point occupies would miss that sign.

Now use a constant radius of one and angle zero. The marked point occupies that same location, one unit to the right, but the instructions are different: aim right and travel one unit forward. A polar location can have more than one coordinate description. The surrounding curve has changed because we changed its rule.

Watch the unit-radius point turn through a complete circuit. This time the distance stays fixed and the result is a circle. Compare that steady distance with the cardioid’s changing one. The geometry comes from what the radius does while the direction turns, not from turning alone.

Return to the cardioid for a final circuit. Read each position in two steps: direction, then signed distance. When the radius is zero, the point is at the origin. When it is negative, reverse the chosen direction. Those instructions are enough to begin reading the many different polar curves in this World.