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Space Curve — Definition, Formula & Examples

A space curve is a curve that exists in three-dimensional space, described by a vector-valued function of a single parameter. Each value of the parameter gives a point (x,y,z)(x, y, z) that the curve passes through.

A space curve is the image of a continuous vector-valued function r:I⊆R→R3\mathbf{r}: I \subseteq \mathbb{R} \to \mathbb{R}^3, where r(t)=⟨x(t), y(t), z(t)⟩\mathbf{r}(t) = \langle x(t),\, y(t),\, z(t) \rangle and xx, yy, zz are continuous functions of the parameter tt on some interval II.

Key Formula

r(t)=⟨x(t), y(t), z(t)⟩\mathbf{r}(t) = \langle x(t),\, y(t),\, z(t) \rangle
Where:
  • r(t)\mathbf{r}(t) = Position vector giving a point on the curve at parameter value t
  • x(t), y(t), z(t)x(t),\, y(t),\, z(t) = Component functions defining the x, y, and z coordinates
  • tt = Parameter, often representing time

How It Works

You define a space curve by specifying three component functions x(t)x(t), y(t)y(t), and z(t)z(t). As tt varies over its domain, the point r(t)\mathbf{r}(t) traces out the curve in 3D. The derivative r′(t)\mathbf{r}'(t) gives the tangent vector at each point, and its magnitude ∣r′(t)∣|\mathbf{r}'(t)| gives the speed of traversal. You can compute arc length, curvature, and torsion from r(t)\mathbf{r}(t) and its derivatives to describe the curve's geometry.

Worked Example

Problem: Find the tangent vector and speed at t=1t = 1 for the space curve r(t)=⟨t2, 2t, t3⟩\mathbf{r}(t) = \langle t^2,\, 2t,\, t^3 \rangle.
Step 1: Differentiate each component with respect to tt.
r′(t)=⟨2t, 2, 3t2⟩\mathbf{r}'(t) = \langle 2t,\, 2,\, 3t^2 \rangle
Step 2: Evaluate the tangent vector at t=1t = 1.
r′(1)=⟨2, 2, 3⟩\mathbf{r}'(1) = \langle 2,\, 2,\, 3 \rangle
Step 3: Compute the speed, which is the magnitude of the tangent vector.
∣r′(1)∣=4+4+9=17|\mathbf{r}'(1)| = \sqrt{4 + 4 + 9} = \sqrt{17}
Answer: The tangent vector at t=1t = 1 is ⟨2,2,3⟩\langle 2, 2, 3 \rangle and the speed is 17\sqrt{17}.

Why It Matters

Space curves appear whenever you model motion in three dimensions — satellite orbits, particle trajectories in physics, and the paths of robotic arms. They are also foundational for computing line integrals and understanding curvature and torsion in multivariable calculus and differential geometry.

Common Mistakes

Mistake: Confusing the parameter tt with one of the spatial coordinates, leading to incorrect derivatives.
Correction: Remember that tt is an independent parameter and xx, yy, zz are all functions of tt. Differentiate each component separately with respect to tt.