Thanks Professor, since previous lecture, I am seeing Bezier control points along with every curves I see around.. B-Spline would be little harder to imagine.. 48:34
The example for geometric continuity is a little confusing (25:00). If the gradient of gamma(t) at t=0 is (0,0), doesn't that make the tangent undefined? If instead we changed the limits such that: gamma(t) = (t,t^2) when *t 0* Then we can instead take the limit of gamma(t) as t approaches zero from the positive direction, i.e. lim |x->0+| T(t) Where T(t) is the tangent of gamma(t) for t > 0: T(t) = gamma'(t)/||gamma'(t)|| = (2t,4t^3)/sqrt(4t^2+16t^6) Therefore: lim |x->0+| T(t) = lim |x->0+| (2t,4t^3)/sqrt(4t^2+16t^6) = lim |x->0+| (1,2t^2)/sqrt(1+4t^2) = (1,0) (The gradient of (t,t^2) at t=0 is (1,0), meaning the tangent is also (1,0). So the tangents match in the way you talk about.) Side point: The "car" doesn't strictly speed up when we pass t=0. It should actually come very close to stopping, before then speeding up considerably. But the main point about the velocity of the car suddenly changing is true. If I am confused or mistaken about any of this, I would love to know. Thanks again for making these lectures publically available.
Thanks Professor, since previous lecture, I am seeing Bezier control points along with every curves I see around.. B-Spline would be little harder to imagine.. 48:34
The example for geometric continuity is a little confusing (25:00).
If the gradient of gamma(t) at t=0 is (0,0), doesn't that make the tangent undefined?
If instead we changed the limits such that:
gamma(t) = (t,t^2) when *t 0*
Then we can instead take the limit of gamma(t) as t approaches zero from the positive direction, i.e. lim |x->0+| T(t)
Where T(t) is the tangent of gamma(t) for t > 0:
T(t) = gamma'(t)/||gamma'(t)|| = (2t,4t^3)/sqrt(4t^2+16t^6)
Therefore:
lim |x->0+| T(t) = lim |x->0+| (2t,4t^3)/sqrt(4t^2+16t^6) = lim |x->0+| (1,2t^2)/sqrt(1+4t^2) = (1,0)
(The gradient of (t,t^2) at t=0 is (1,0), meaning the tangent is also (1,0). So the tangents match in the way you talk about.)
Side point: The "car" doesn't strictly speed up when we pass t=0. It should actually come very close to stopping, before then speeding up considerably. But the main point about the velocity of the car suddenly changing is true.
If I am confused or mistaken about any of this, I would love to know. Thanks again for making these lectures publically available.
Come on, prof! SimCity 2000 was *reticulating* splines. Anything else would make no sense!
what is this 'alpha' symbol in 18:23 is it directly proportional? or denoting a cross product.
directly proportional