Why is the red part true?
From: http://graphics.stanford.edu/courses/cs348a-17-winter/Papers/quaternion.pdf
2026-04-04 16:35:25.1775320525
Reparameterizing quaternion in terms of angle
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HINT. Notice $\sin ^2\theta+\cos^2\theta=1$. What does $\theta\in [0,\pi]$ lead to? Consider its graph $(x=\cos \theta, y=\sin \theta )$ on the plane, with $x^2+y^2=1$, which is exactly an upper semicircle.