3 ms·
Bezier curves. Cubic bezier curves (generally the most common type) are represented by cubic polynomials x(t) and y(t) (or just one polynomial P(t) where the c
by armchairhacker 4y ago
Bezier curves.
Cubic bezier curves (generally the most common type) are represented by cubic polynomials x(t) and y(t) (or just one polynomial P(t) where the coefficients are vectors). The first derivative gives you the tangent and normal. The second derivative gives you the direction the normal will gradually become as t increases. The third derivative gives you the change in this change (the fourth derivative is zero). Curvature is also computed from the first and second derivatives: K = det(P', P'')/||P'||^3
Other kinds of bezier curves are similar. In quadratic ones the first and second derivatives are normal and normal change but the third derivative is zero. In higher-polynomial bezier curves the third derivative changes over time as the fourth is non-zero, and so on.
Source is a really interesting video on bezier curves which I highly recommend: https://www.youtube.com/watch?v=aVwxzDHniEw https://www.youtube.com/watch?v=aVwxzDHniEw.
- _nhynes 4y agoAnd these are common in everyday life due to screensavers, right?
- krallja 4y agoFonts, vector graphics, interpolation for animation and video games
- Someone 4y agoI think that, nowadays, it’s more likely that fonts use quadratic Bézier curves (as in TrueType and its derivatives), rather than cubic ones (as in PostScript and Adobe’s Type 1 fonts), but since OpenType supports both (https://en.wikipedia.org/wiki/OpenType#Description https://en.wikipedia.org/wiki/OpenType#Description) I may be wrong. Similarly, SVG supports both quadratic and cubic Bézier curves.
- iainmerrick 4y agoAnd cars. Curves with higher-order continuity is why modern cars have such interesting smoothly curved surfaces.
- psyc 4y agoThis was the raison d'etre for Bézier curves. Its inventor and re-inventor worked at car companies. https://en.wikipedia.org/wiki/B%C3%A9zier_curve#Invention https://en.wikipedia.org/wiki/B%C3%A9zier_curve#Invention
- lloeki 4y agoRoads and tracks. Connecting straightaways and turns to at least second order (sometimes third, esp. for high speed trains) makes transitions much smoother as it reduces or eliminates jerk. https://en.m.wikipedia.org/wiki/Jerk_(physics) https://en.m.wikipedia.org/wiki/Jerk_(physics) Even during driving, intuitively you don't yank the steering wheel to the desired position but produce a eased-in eased-out progressive movement that results in such a trajectory.