HackerThe path looks smooth. Why does the camera still jerk?
Follow two complementary explanations by Freya Holmér: first inspect how progress along one curve is measured, then how two curves join. Constant speed, continuous tangents and continuous curvature are different requirements.
Questions in this reading · 3
A parameter percentage is not a distance percentage
Start with the moving dots in The Beauty of Bézier Curves. Equal increments of t do not guarantee equal distances, so motion can speed up and slow down even within one unbroken curve. Holmér samples accumulated length, looks up t for a desired distance, and evaluates the original curve. This changes traversal rather than shape. It approximates constant speed within a segment, but does not by itself guarantee matching direction or turning behavior where another segment begins. The two videos address complementary questions: how to travel along a curve, and how curves connect.
Passages and timestamps · 2
equally spaced t values are not equally spaced by distancegenerate what's called a lookup tableNo visible corner does not mean continuous velocity
In The Continuity of Splines, aligned handles make the outline look smooth, yet the moving boxes change speed at the join. Alignment constrains direction, giving G¹ continuity. Under her uniform parameter intervals, mirroring the handles also matches their lengths and the first derivatives, giving C¹ continuity. Continuous velocity is not constant velocity: motion may gradually accelerate without a sudden jump. The acceleration arrows can still jump on a C¹ path, so camera motion may require a C² check. These conditions concern the chosen parameterization as well as the outline; a still image alone cannot establish them.
Passages and timestamps · 1
So even though we aligned our tangents, there’s a sudden change in speedReflections ask a different smoothness question
For a reflective shell, traversal speed is no longer the issue. Holmér joins a circular arc tangentially to a straight line: direction matches, but the reflection still has a seam because the arc has curvature while the line has none. Matching curvature, giving G², fixes the demonstrated reflection. Keep three checks separate: a distance table for traversal, velocity and acceleration graphs for parameterized motion, and a curvature comb for geometry. They are not one score to maximize; interpolation through control points and local editing also matter. These are two mathematical explanations by the same author, forming a learning sequence. Implications from parametric to geometric continuity retain the nonzero-velocity condition.
Passages and timestamps · 2
we also needed curvature continuityequally spaced t values are not equally spaced by distanceTry it on your own example
Inspect a camera path twice: sample equal times within each segment and compare distances, then compare velocity and acceleration on either side of each join. For a reflective surface, inspect tangents and curvature instead. Choose the quantity that must be continuous before choosing the spline and parameterization.