Sinusoidal Curves in Three Dimensions

Building Space Curves from Sinusoids

A space curve can be designed by specifying how each coordinate changes with a parameter. If those coordinates vary sinusoidally, the resulting point moves through space in a repeated wave-like pattern. Changing the amplitude, phase, or frequency of one coordinate changes how that motion is arranged.

A sinusoidal coordinate does not have to be accompanied by another sinusoid. A curve may have two varying coordinates and one constant coordinate, or all three coordinates may vary. The key is to examine each component and then check how the components act together.

Definition: A sinusoidal parametrization in three dimensions has the form \(\mathbf{r}(t)=\langle c_1+A_1\cos(\omega_1t+\phi_1),\ c_2+A_2\cos(\omega_2t+\phi_2),\ c_3+A_3\cos(\omega_3t+\phi_3)\rangle\), or the corresponding form using sine in any component. Here \(t\) lies in a specified parameter interval, \(c_i\) is a coordinate offset, \(A_i\geq 0\) is an amplitude, \(\omega_i\) is an angular frequency, and \(\phi_i\) is a phase. A component can be constant, for example when \(A_i=0\).

The offsets \(c_i\) set the midlines of the coordinate oscillations. The amplitude measures the distance from a midline to an extreme value. When \(A_i>0\) and \(\omega_i\neq 0\), the coordinate varies between \(c_i-A_i\) and \(c_i+A_i\), and its least positive period is \(2\pi/|\omega_i|\). The phase determines where in its oscillation the coordinate is when \(t=0\).

Frequency describes how rapidly a coordinate completes its oscillations as \(t\) increases. A larger absolute frequency means a shorter coordinate period. If \(\omega_i=0\), the angle in that coordinate is fixed, so the coordinate is constant. If \(A_i=0\), the coordinate is also constant, whatever value \(\omega_i\) has. In either constant case every positive increment is a period, and there is no least positive period.

Read Each Coordinate Before the Whole Curve

For a component \(q(t)=c+A\cos(\omega t+\phi)\), the range is contained in \([c-A,c+A]\), because the cosine always lies between \(-1\) and \(1\). If \(A>0\) and \(\omega\neq0\), it completes one full cycle whenever its angle changes by \(2\pi\). Thus the least positive period is \(2\pi/|\omega|\). The same conclusions hold for a sine component.

These facts describe individual coordinates, not automatically the whole space curve. A vector-valued function repeats after an increment \(T>0\) only if all its coordinates repeat after that same increment. Phases affect the locations along the parameter axis where particular values occur, but do not change the period of a nonconstant coordinate.

Worked Example: Two Oscillating Coordinates and One Constant Coordinate

Consider \(\mathbf{r}(t)=\langle 3+2\cos t,\ -1+\sin t,\ 4\rangle\), for \(0\leq t\leq 2\pi\). The first coordinate has offset \(3\), amplitude \(2\), and frequency \(1\), so its values lie in \([1,5]\). The second has offset \(-1\), amplitude \(1\), and frequency \(1\), so its values lie in \([-2,0]\). The third coordinate is constantly \(4\).

At \(t=0\), \(\cos 0=1\) and \(\sin 0=0\), giving \(\mathbf{r}(0)=\langle 3+2(1),-1+0,4\rangle=\langle 5,-1,4\rangle\).

At \(t=\pi/2\), \(\cos(\pi/2)=0\) and \(\sin(\pi/2)=1\), giving \(\mathbf{r}(\pi/2)=\langle 3,-1+1,4\rangle=\langle 3,0,4\rangle\).

At \(t=\pi\), \(\cos\pi=-1\) and \(\sin\pi=0\), so \(\mathbf{r}(\pi)=\langle 3-2,-1,4\rangle=\langle 1,-1,4\rangle\).

At \(t=3\pi/2\), \(\cos(3\pi/2)=0\) and \(\sin(3\pi/2)=-1\), so \(\mathbf{r}(3\pi/2)=\langle 3,-2,4\rangle\). Finally, at \(t=2\pi\), the cosine and sine return to \(1\) and \(0\), so \(\mathbf{r}(2\pi)=\langle 5,-1,4\rangle=\mathbf{r}(0)\). This checks the coordinate ranges at their extremes and confirms that the parametrization returns to its starting point after one cycle.

Amplitude, Phase, and Frequency in Combination

It is useful to change one feature at a time when comparing sinusoidal curves. Increasing an amplitude increases the coordinate’s range about its offset. Changing a phase shifts the timing of that coordinate’s oscillation. Changing a frequency changes its period and therefore how often it oscillates while the parameter runs through a fixed interval.

A phase change in one coordinate need not be a simple shift of the whole space curve. A common shift of the parameter changes every component in a coordinated way: replacing \(t\) by \(t+\tau\) changes the phase of component \(i\) by \(\omega_i\tau\). If only one component’s phase changes, the relative timing among coordinates can change, and the trace may change as well.

Worked Example: Comparing Phase and Frequency

Take \(\mathbf{r}(t)=\langle 2\cos(2t+\pi/2),\ 3\sin(t-\pi/2),\ \cos(2t)\rangle\). The coordinate amplitudes are \(2\), \(3\), and \(1\). Their angular frequencies are \(2\), \(1\), and \(2\), respectively. Thus the first and third coordinates have least positive period \(2\pi/2=\pi\), while the second has least positive period \(2\pi\). The coordinate ranges are \([-2,2]\), \([-3,3]\), and \([-1,1]\).

At \(t=0\), the angles are \(\pi/2\), \(-\pi/2\), and \(0\). Therefore \(\mathbf{r}(0)=\langle 2\cos(\pi/2),3\sin(-\pi/2),\cos 0\rangle=\langle 0,-3,1\rangle\).

At \(t=\pi/4\), the angles are \(\pi\), \(-\pi/4\), and \(\pi/2\). Hence \(\mathbf{r}(\pi/4)=\langle 2\cos\pi,3\sin(-\pi/4),\cos(\pi/2)\rangle=\langle -2,-3\sqrt{2}/2,0\rangle\).

At \(t=\pi/2\), the angles are \(3\pi/2\), \(0\), and \(\pi\), so \(\mathbf{r}(\pi/2)=\langle 0,0,-1\rangle\). The first and third coordinates have completed half of their periods, while the second has completed one quarter of its period.

To check a shared period, advance the parameter by \(2\pi\). The angle increases are \(4\pi\), \(2\pi\), and \(4\pi\), all whole-number multiples of \(2\pi\). Every coordinate therefore returns to its original value, and \(\mathbf{r}(t+2\pi)=\mathbf{r}(t)\). The differing frequencies make the coordinates oscillate at different rates, even though their combined parametrization repeats.

When Do the Coordinates Share a Period?

Suppose there are finitely many nonconstant sinusoidal coordinates, with nonzero angular frequencies \(\omega_1,\ldots,\omega_n\). Their individual least positive periods are \(p_i=2\pi/|\omega_i|\). A positive number \(T\) is a period of the entire parametrization precisely when it is a period of every nonconstant coordinate. Equivalently, for each \(i\), \(T/p_i\) must be a positive integer.

Theorem: For finitely many nonconstant sinusoidal coordinates, a common positive period exists if and only if the ratio of every pair of coordinate periods is rational. Constant coordinates impose no restriction on a common period.

Justification. If \(T\) is a common period, then for each coordinate \(T=m_i p_i\) for some positive integer \(m_i\). For any pair \(i,j\), this gives \(p_i/p_j=m_j/m_i\), a rational number. Conversely, if all period ratios are rational, write each period as a rational multiple of one fixed period \(p_1\). Write \(p_i/p_1=a_i/b_i\) in lowest terms with positive integers \(a_i,b_i\). Let \(N\) be a common multiple of all the numerators \(a_i\), and set \(T=Np_1\). Then \(T/p_i=Nb_i/a_i\) is an integer for every \(i\), so \(T\) is an integer multiple of each \(p_i\). That value is a common period. Constant coordinates repeat for every increment and so cannot prevent one from being a common period.

Worked Example: Finding the Shared Period

Consider \(\mathbf{r}(t)=\langle \cos(2t),\ \sin(3t),\ 2\cos(4t+\pi/3)\rangle\). The coordinate periods are \(2\pi/2=\pi\), \(2\pi/3\), and \(2\pi/4=\pi/2\). A common period must be an integer multiple of each.

The value \(2\pi\) works: it is \(2\) times \(\pi\), \(3\) times \(2\pi/3\), and \(4\) times \(\pi/2\). For example, at \(t=0\), \(\mathbf{r}(0)=\langle 1,0,2\cos(\pi/3)\rangle=\langle 1,0,1\rangle\). At \(t=2\pi\), the angles become \(4\pi\), \(6\pi\), and \(8\pi+\pi/3\). Their cosine or sine values match those at \(t=0\), so \(\mathbf{r}(2\pi)=\langle 1,0,1\rangle=\mathbf{r}(0)\).

To see that no smaller positive common period works, suppose \(T\) is one. The first coordinate requires \(T=m\pi\) for a positive integer \(m\), and the second requires \(T=2n\pi/3\) for a positive integer \(n\). Equating these gives \(3m=2n\). Consequently \(m\) must be even, so \(T\geq2\pi\). Since \(2\pi\) also works for the third coordinate, it is the least positive period of the full parametrization.

A Common Period Mistake

A common error is to find the period of one coordinate and assume that it is automatically a period of the vector-valued function. For example, in the preceding parametrization the first coordinate has period \(\pi\). But the second coordinate does not: at \(t=\pi/6\), its value is \(\sin(\pi/2)=1\), whereas at \(t=\pi/6+\pi=7\pi/6\), its value is \(\sin(7\pi/2)=-1\). Thus \(\pi\) does not repeat the full parametrization; the common period is \(2\pi\).

There is a separate constant-coordinate caution. If a component has amplitude zero, it is constant and every positive increment is a period; it does not have a least positive period. The formula \(2\pi/|\omega|\) for a least positive period applies only when the amplitude is positive and the frequency is nonzero.

Why These Curves Matter

Sinusoidal coordinate functions provide a direct way to design and interpret periodic motion in space. They let you specify the range of each coordinate, the timing of its peaks and troughs, and how quickly it oscillates. Comparing these components helps explain why two curves with similar coordinate ranges can nevertheless trace different paths: their coordinates may reach corresponding values at different parameter times.

When analyzing a proposed parametrization, first identify the amplitude, offset, phase, and frequency of each varying component. Then evaluate a few parameter values to understand how the components line up. Finally, test any claimed period in every coordinate. That order keeps coordinate-level facts distinct from conclusions about the full space curve.

Check Your Understanding

Use the coordinate functions and their shared behavior to answer these questions.

For \(q(t)=5+3\cos(4t-\pi/2)\), what are the offset, amplitude, range, and least positive period?
If a sinusoidal coordinate has amplitude zero, does it have a least positive period? Explain.
For a vector-valued parametrization with coordinate periods \(2\pi\) and \(\pi/3\), give one common positive period.
Why does changing the phase of just one coordinate not necessarily amount to shifting the parameter for the whole curve?
What condition must a positive increment satisfy to be a period of a space-curve parametrization?