exp: Lie algebra → Lie group — tangent vectors generate group elements
The exponential map turns a direction at the identity into a point of the group: pick a velocity, flow for one unit of time, and see where you land.
Adjust the angular velocity and watch the glowing particle flow from the tangent line onto the circle, wrapping around once the angle passes 2π.
The exponential map is the bridge between a Lie algebra (the tangent space at the identity) and the Lie group itself. Given a tangent vector X, the exponential exp(X) “flows” along the group for unit time, producing a group element. For matrix groups, this is literally the matrix exponential: exp(X) = I + X + X²/2! + X³/3! + …
On SO(2), the Lie algebra is just ℝ (angular velocities). The exponential map sends ω to rotation by angle ω. Adjust the angular velocity and watch a glowing particle flow from the tangent line along the circle, wrapping around as the angle increases past 2π.
Key insight: The exponential map for SO(2) is surjective — every rotation is reachable. It wraps the real line (the algebra) around the circle (the group) like thread on a spool.
Drag the algebra vector and watch the family of rotations it generates traced through the SO(3) ball, with trails on the rotating frame.
In SO(3), the Lie algebra so(3) consists of 3×3 skew-symmetric matrices, parameterized by ℝ³. Each algebra element ω generates a one-parameter subgroup: a family of rotations around the axis ω/|ω|. Drag the algebra vector to see the corresponding one-parameter subgroup traced on the SO(3) ball, with afterimage trails on the rotating frame.
Add terms of the series one at a time. Each term gets its own color, and the transformed grid closes in on the exact exponential.
The matrix exponential exp(X) = I + X + X²/2! + X³/3! + … is an infinite series that always converges. Watch the partial sums converge as terms are added: each order gets a spectral color, and the grid transformation approaches the exact exponential.
Click to place a target rotation and see its logarithm. Move toward rotation by π and find the glowing seam where two answers compete.
Near the identity, the exponential map is a diffeomorphism — it gives a perfect local coordinate system. But globally it can fail to be surjective: some group elements may not be reachable by a single exponential. Understanding where and why this fails reveals deep structure in the group.
The matrix logarithm is the inverse of the exponential — when it exists. Near the identity, log is well-defined and smooth. But near rotation by π, the logarithm becomes multi-valued: two candidate algebra elements map to the same group element. Click to place a target rotation and see the logarithm computed. The “fold” near π glows as a seam on the ball surface.
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