Rodrigues Rotation Formula

Rodrigues’ rotation formula gives a closed-form expression for rotating a 3D vector around an arbitrary axis using basic vector operations: dot product, cross product, and scalar multiplication. It expresses the rotated vector $$\mathbf{v}$_{rot}$ as a sum of three geometrically meaningful components: the original vector scaled by $\cos$\theta$$, a perpendicular component obtained via the cross product scaled by $\sin$\theta$$, and a projection of $$\mathbf{v}$$ onto the axis scaled by $1 - \cos$\theta$$. This avoids building a full $3\times3$ rotation matrix explicitly, while being mathematically equivalent to a rotation in $SO(3)$.

Geometrically, any vector $$\mathbf{v}$$ can be decomposed into a part parallel to the rotation axis $$\mathbf{k}$$ and a part perpendicular to it. The parallel component is unchanged by rotation, while the perpendicular component rotates in the plane orthogonal to $$\mathbf{k}$$ by angle $$\theta$$. In computer vision and robotics, this axis–angle parameterization is widely used to represent camera orientation, perform pose updates, and convert between rotation matrices and more compact representations.

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1. Background Knowledge (Key Concepts)

• Axis–angle representation A 3D rotation can be represented by:

• A unit axis $$\mathbf{k}$\in \mathbb{R}^3$ (direction of rotation)

• A rotation angle $$\theta$$ (right-hand rule about $$\mathbf{k}$$) Rodrigues’ formula maps this pair $($\mathbf{k}$,$\theta$)$ and a vector $$\mathbf{v}$$ to the rotated vector $$\mathbf{v}$_{rot}$.

• Vector operations involved

• Dot product $$\mathbf{k}$\cdot$\mathbf{v}$$: projection of $$\mathbf{v}$$ on the axis.

• Cross product $$\mathbf{k}$\times$\mathbf{v}$$: vector orthogonal to both $$\mathbf{k}$$ and $$\mathbf{v}$$, giving the direction of infinitesimal rotation.

• Decomposition: $$\mathbf{v}$=$\mathbf{v}$_\parallel +$\mathbf{v}$_\perp$ where $$\mathbf{v}$_\parallel = ($\mathbf{k}$\cdot$\mathbf{v}$)$\mathbf{k}$$, $$\mathbf{v}$_\perp =$\mathbf{v}$-$\mathbf{v}$_\parallel$. Only $$\mathbf{v}$_\perp$ actually “spins” in the rotation.

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2. Algorithm / General Approach

You are given:

• A unit axis $$\mathbf{k}$= (k_x, k_y, k_z)$
• An angle $$\theta$$
• A 3D point/vector $$\mathbf{v}$= (x, y, z)$

You need to compute:

$$\mathbf{v}_{rot} = \mathbf{v}\cos\theta + (\mathbf{k} \times \mathbf{v})\sin\theta + \mathbf{k}(\mathbf{k} \cdot \mathbf{v})(1 - \cos\theta)$$

The algorithm pattern is:

1. Precompute $\cos$\theta$$ and $\sin$\theta$$.
2. Compute the dot and cross of $$\mathbf{k}$$ and $$\mathbf{v}$$.
3. Form each term of the formula with scalar–vector multiplies.
4. Sum the three terms to get the rotated vector.

This is a direct formula evaluation (constant time, no loops).

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3. Step-by-Step Strategy to Implement

Assume inputs:

• k = (kx, ky, kz) (axis, should be unit)
• theta (angle, in radians)
• v = (vx, vy, vz) (point or direction)

1. (Optionally) normalize the axis If the problem does not guarantee a unit axis, normalize:

norm = sqrt(kx*kx + ky*ky + kz*kz)
kx, ky, kz = kx/norm, ky/norm, kz/norm

2. Precompute trigonometric values

c = cos(theta) # cosθ
s = sin(theta) # sinθ
one_minus_c = 1 - c

3. Compute dot product $$\mathbf{k}$\cdot$\mathbf{v}$$

kv = kx*vx + ky*vy + kz*vz # scalar

4. Compute cross product $$\mathbf{k}$\times$\mathbf{v}$$

cx = ky*vz - kz*vy
cy = kz*vx - kx*vz
cz = kx*vy - ky*vx

5. Assemble the three terms of Rodrigues’ formula

• First term: $$\mathbf{v}$\cos$\theta$$

term1x = vx * c
term1y = vy * c
term1z = vz * c

• Second term: $($\mathbf{k}$\times$\mathbf{v}$)\sin$\theta$$

term2x = cx * s
term2y = cy * s
term2z = cz * s

• Third term: $$\mathbf{k}$($\mathbf{k}$\cdot$\mathbf{v}$)(1 - \cos$\theta$)$

scale = kv * one_minus_c
term3x = kx * scale
term3y = ky * scale
term3z = kz * scale

6. Sum terms to get the rotated vector

vrot_x = term1x + term2x + term3x
vrot_y = term1y + term2y + term3y
vrot_z = term1z + term2z + term3z

7. Return $(vrot\_x, vrot\_y, vrot\_z)$.

This is the full computation pattern; details like language syntax differ but the operations are identical.

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4. Common Pitfalls

• Axis not normalized Rodrigues’ formula assumes $\|$\mathbf{k}$\| = 1$. If it is not, the rotation will scale or distort the vector. Either:

• Normalize k yourself, or

• Trust the problem’s guarantee and avoid double-normalizing.

• Degrees vs radians Most math libraries expect radians. If you are given degrees, convert:

theta_rad = theta_deg * pi / 180.0

• Incorrect cross product order The formula uses $$\mathbf{k}$\times$\mathbf{v}$$, not $$\mathbf{v}$\times$\mathbf{k}$$. Reversing order flips the sign of the rotation (clockwise vs counterclockwise).

• Floating point precision For very small $$\theta$$, sin(theta) and 1 - cos(theta) can become numerically delicate. Using standard sin and cos is usually fine, but be aware of rounding errors if you later rely on length being exactly preserved.

• Confusing point vs vector Rodrigues’ formula applies to vectors from the origin. If you are rotating a point around an axis not passing through the origin, you must:

• Translate the point so that the axis passes through the origin.

• Apply Rodrigues’ formula.

• Translate back.

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5. Time & Space Complexity

• Time complexity The computation is a constant number of scalar operations (adds, multiplies, trig calls, dot/cross products), so:

• Time: $O(1)$ per vector.

• Space complexity

• You store a constant number of scalars and vectors (no arrays growing with input size):

• Space: $O(1)$.

This makes Rodrigues’ formula very efficient for repeated 3D rotations in vision, graphics, and robotics.