Lecture 7: Intro to Geometry, Splines, and Bezier Curves (34)

TonyLianLong

We want to match the derivative because we want the curve to be c1 continuous. Is this right?

Sicheng-Pan

Actually I think we want to match the derivative because we hope that the derivative of the combined curve is continuous. The combined curve itself is continuous if we match the end points of the segments.

saltyminty

Could we manipulate the second derivative(s) of x to achieve a similar effect to extending the handlebar/control points in bezier curves? Ie a point having a much smaller second derivative to mimic the handle bar being very long/the control point being very far, thus causing it to curve less. Is it possible to set up the equations that way?

StaffDanCubed

@Sicheng-Pan yes I think so too. Another way to think about is that geometrically, we want the curve to be tangent to the given lines at the two end points so that it’ll smoothly transition into the next curve (which will be tangent to the same line). Mathematically, tangent=same slopes=same derivative values.

StaffDanCubed

@saltyminty I’m not 100% sure but from my understanding so far, this doesn’t seem possible because the 4 equations shown here can uniquely determine a cubic function, which has 4 unknowns. If you want to further adjust, you might have to increase the degree of the polynomial.

Staffjamesfobrien

If you increase the polynomial to degree 5 then you have 2 more coefficients so you can add 2 more constraints, for example the 2nd derivatives as u0 and u1.

We want to match the derivative because we want the curve to be c1 continuous. Is this right?

Actually I think we want to match the derivative because we hope that the derivative of the combined curve is continuous. The combined curve itself is continuous if we match the end points of the segments.

Could we manipulate the second derivative(s) of x to achieve a similar effect to extending the handlebar/control points in bezier curves? Ie a point having a much smaller second derivative to mimic the handle bar being very long/the control point being very far, thus causing it to curve less. Is it possible to set up the equations that way?

@Sicheng-Pan yes I think so too. Another way to think about is that geometrically, we want the curve to be tangent to the given lines at the two end points so that it’ll smoothly transition into the next curve (which will be tangent to the same line). Mathematically, tangent=same slopes=same derivative values.

@saltyminty I’m not 100% sure but from my understanding so far, this doesn’t seem possible because the 4 equations shown here can uniquely determine a cubic function, which has 4 unknowns. If you want to further adjust, you might have to increase the degree of the polynomial.

If you increase the polynomial to degree 5 then you have 2 more coefficients so you can add 2 more constraints, for example the 2nd derivatives as u0 and u1.