Circles
In a Euclidean plane, fix a point \(c\) and a positive real number \(r\). The circle with center \(c\) and radius \(r\) is the set of points at distance \(r\) from \(c\).
To describe a circle algebraically, we might use Cartesian coordinates. Write \(c = (x_{0}, y_{0})\), and let \((x, y)\) denote the coordinates of a general point on the circle. The line segment with endpoints \((x_{0}, y_{0})\) and \((x, y)\) may be viewed as the hypotenuse of a right triangle with sides \(|x - x_{0}|\) and \(|y - y_{0}|\), the horizontal and vertical separations from the center to \((x, y)\). By the Pythagorean theorem, the sum of the squares of the sides is the square of the hypotenuse: \[ |x - x_{0}|^{2} + |y - y_{0}|^{2} = r^{2}. \] Since \(|u|^{2} = u^{2}\) for every real number \(u\), the preceding equation may be written \[ (x - x_{0})^{2} + (y - y_{0})^{2} = r^{2}. \] A point \((x, y)\) of the plane lies on the circle of center \(c\) and radius \(r\) (geometry) if and only if the preceding equation is true (algebra).
The unit circle is the circle centered at the origin, \(c = (0, 0)\), and of radius \(r = 1\). Its Cartesian equation is \[ x^{2} + y^{2} = 1. \]
Angle
Take a vertical line segment of length \(\ell\), place its bottom end at the point \((1, 0)\) on the unit circle, and ``wrap'' the segment smoothly counterclockwise around the unit circle. The top end of the segment ``maps to'' a well-defined point \(p(\ell)\) of the circle. This function notation reminds us that the length \(\ell\) uniquely defines a point on the circle.
There is a positive real number \(2\pi \approx 6.283\) characterized by two properties:
- The segment of length \(2\pi\) wraps around: The upper endpoint maps to \((1, 0)\).
- No smaller positive real number \(\ell\) wraps around the circle.
Algebraically, \(p(2\pi) = (1, 0)\), and if \(0 < \ell < 2\pi\) then \(p(\ell) \neq (1, 0)\).
If \(\ell\) is negative, we place the top end of the segment at \((1, 0)\), and define \(p(\ell)\) by wrapping the segment clockwise around the circle.
For each real number \(\ell\), we say \(p(\ell)\) makes angle \(\ell\) with \(p(0) = (1, 0)\). A point on the circle does not have a unique angle. For example, \(p(0) = (1, 0) = p(2\pi)\), so the point \((1, 0)\) has angles \(0\), and \(2\pi\), and also \(-2\pi\), \(4\pi\), and so on. Generally, each point has infinitely many distinct angles, any two differing by a whole multiple of \(2\pi\). (Mathematicians prefer to measure angles in radians rather than degrees because formulas involving calculus come out more neatly.)
For cultural reasons, angle is not denoted by \(\ell\), but by the Greek letter \(\theta\) (theta). If \(\theta\) is real, we write \(p(\theta)\) for the point on the unit circle making angle \(\theta\) with \((1, 0)\).
If \(p_{1} = p(\theta_{1})\) and \(p_{2} = p(\theta_{2})\) are points of the unit circle, we would like to view \(|\theta_{2} - \theta_{1}|\) as a measure of ``distance'' between \(p_{1}\) and \(p_{2}\). Because \(\theta_{1}\) and \(\theta_{2}\) are not uniquely defined by the points, however, our numerical quantity is not well-defined.
To remedy this, we focus our attention on the smallest distance over all choices of \(\theta_{1}\) and \(\theta_{2}\). The angular separation of two points \(p_{1}\) and \(p_{2}\) on the unit circle is the smallest value of \(|\theta_{2} - \theta_{1}|\) subject to \(p_{1} = p(\theta_{1})\) and \(p_{2} = p(\theta_{2})\).
The angular separation is non-negative, as the minimum of a collection of non-negative real numbers. Further, the angular separation of two points is \(0\) if and only if the points are the same. Finally, the angular separation is never larger than \(\pi\), one-half the circumference, and is equal to \(\pi\) if and only if the points are opposite ends of a diameter. Intuitively, any two points on the unit circle lie in some half-circle, and so are separated by a distance ``along the circle'' of at most half the circumference, i.e., \(\pi\).
Trigonometry
For each real \(\theta\), there is a unique point \(p(\theta)\) on the unit circle making angle \(\theta\) with \((1, 0)\). The horizontal and vertical coordinates of \(p(\theta)\) are the cosine of \(\theta\), denoted \(\cos\theta\), and the sine of \(\theta\), denoted \(\sin\theta\). We call \(\cos\) and \(\sin\) the circular functions.
Briefly, \(p(\theta) = (\cos\theta, \sin\theta)\).
The Pythagorean theorem, which gives the Cartesian equation \(x^{2} + y^{2} = 1\) for the unit circle, tells us \((\cos\theta)^{2} + (\sin\theta)^{2} = 1\). Mathematicians write \(\cos^{2}\) and \(\sin^{2}\) for the squares of the cosine and sine, and therefore write this basic identity \(\cos^{2} \theta + \sin^{2} \theta = 1\).
For each real \(\theta\), we have \(\cos(\theta + 2\pi) = \cos\theta\) and \(\sin(\theta + 2\pi) = \sin\theta\). We say the circular functions are periodic with period \(2\pi\), or \(2\pi\)-periodic. Geometrically, their graphs ``repeat'' after \(2\pi\).
Rotation
There is a geometric operation of ``rotation'' about the origin of the Euclidean plane that fixes (does not move) the origin, preserves Euclidean distances, and ``preserves relative angles'' at the origin. For each real number \(\theta\), rotation by \(\theta\) about the origin maps the rightmost point \((1, 0)\) of the unit circle to the point \(p(\theta) = (\cos\theta, \sin\theta)\).
Rotation by \(0\) is the ``identity map'' on the plane: Each point maps to itself. Because an angle \(\theta = 2\pi\) represents one full turn, rotation by \(2\pi\) is also the identity map: If we rotate the plane one full turn about the origin, every point maps to itself. The same is true of rotation by every whole multiple of \(2\pi\).
Rotations satisfy a remarkable composition law. For example, if we rotate the plane by an angle \(\frac{\pi}{2}\) (\(90\) degrees) and then by angle \(\frac{\pi}{3}\) (\(60\) degrees), the net result is rotation by the sum of the angles, \(\frac{\pi}{2} + \frac{\pi}{3} = \frac{5\pi}{6}\) (\(150\) degrees).
Generally, if we rotate about the origin by an angle \(\theta_{1}\), and then rotate about the origin by an angle \(\theta_{2}\), the net result is rotation about the origin by the sum, \(\theta_{1} + \theta_{2}\).
We might express the composition law for rotations in symbols by letting \(R_{\theta}\) denote the operation of rotating about the origin by angle \(\theta\). This entity \(R_{\theta}\) is a type of function. Its inputs and outputs are points of the plane. If \((x, y)\) is a point of the plane, then \(R_{\theta}(x, y)\) denotes the image of \((x, y)\) under rotation.
If we write \((x', y') = R_{\theta_{1}}(x, y)\), giving a name to the ``output'' of \(R_{\theta_{1}}\) acting on \((x, y)\), substitution tells us that \[ R_{\theta_{2}}(x', y') = R_{\theta_{2}}\bigl(R_{\theta_{1}}(x, y)\bigr) \] is the image of the point \(R_{\theta_{1}}(x, y)\) under \(R_{\theta_{2}}\), namely the net result of rotating \((x, y)\) about the origin by angle \(\theta_{1}\) and then by \(\theta_{2}\).
The composition law would then be written \[ R_{\theta_{1} + \theta_{2}}(x, y) = R_{\theta_{2}}\bigl(R_{\theta_{1}}(x, y)\bigr). \] There is one further abbreviation we can make. The preceding relation holds for every point \((x, y)\), so mentioning \((x, y)\) is unnecessary. Abstracting to the operations themselves, we might write \[ R_{\theta_{1} + \theta_{2}} = R_{\theta_{2}} R_{\theta_{1}}. \] In symbols, this says ``the result of rotating about the origin by angle \(\theta_{1}\) and then by angle \(\theta_{2}\) has the same effect as the single rotation through angle \(\theta_{1} + \theta_{2}\).
Groups
In particular, the composition law for rotations says a composition of rotations about the origin is a rotation about the origin. Mathematicians say the set of rotations about the origin is ``closed under composition.''
Further, the identity mapping is a rotation (by angle \(0\), or any whole multiple of \(2\pi\)), and the ``inverse'' of a rotation about the origin—the operation that ``undoes'' the rotation—is itself a rotation about the origin.
These three properties—a set \(G\) of mappings that is closed under composition, contains the identity map, and contains the inverse of each mapping in \(G\)—characterize a mathematical structure known as a group. The group of plane rotations about the origin is a personal favorite at Differential Geometry, and underlies many images and objects in our math art shop.
If we fix an arbitrary point \(p\) of the plane, the set of rotations about \(p\) is also a group under composition. In a mathematical sense (that we will not make precise here), this group of rotations has the same abstract structure as the group of rotations about the origin. Mathematicians say any two of these groups are isomorphic (iso: same, morph: form).
The set of rotations about arbitrary points of the plane, by contrast, is not a group. Can you see why? (Caution: Unless centers and angles are chosen carefully, a composition of rotations with different centers is a rotation, although its center is not the center of either ``factor.'')
The identity mapping is a particularly special rotation: By itself it forms a group! No other single rotation has this property.
Among non-identity rotations, there is the interesting class of half-turns, rotations by angle \(\pi\) (or \(180\) degrees) about some point. The set of all half-turns does not form a group. (Hint: What is the effect of successively performing two half-turns with different centers?) The identity mapping together with a half-turn a particular point does form a group, with just two elements.
If we focus on the half-turn about the origin, our group is abstractly the set \(\{1, -1\}\) under multiplication. The identity mapping sends each point \((x, y)\) to \(1 \cdot (x, y) = (x, y)\). The half-turn about the origin sends each point \((x, y)\) to \((-1) \cdot (x, y) = (-x, -y)\). Composition effects multiplication of ``coefficients.''
One final observation is worth making: Consider a quarter-turn about the origin. (There are two, counterclockwise and clockwise.) Performing the same quarter-turn twice gives a half-turn. If there were a way to view a quarter-turn about the origin as ``multiplication'' by some kind of number, then the square of that number would be the coefficient of a half-turn, namely \(-1\). In other words, we would have a concrete implementation of the (sometimes mysterious) ``square root of \(-1\).'' This observation is a departure point for the world of complex numbers.