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apatriarca

Member Since 03 Jul 2006
Offline Last Active Today, 09:41 AM

Posts I've Made

In Topic: Editing Bezier Splines while maintaining C1 Continuity

23 June 2014 - 01:46 AM

To maintain C1 continuity between consecutive (cubic) Bézier splines you have to maintain the following equations:

 

P3(i-1) - P2(-1) = P1(i) - P0(i)

P3(i-1) = P0(i)

 

where the index in the parenthesis represents the index of the curve in the spline. The derivative/tangent at the beginning (end) of Bézier curve is in fact parallel to the edge between the first two (last two) control points. If the two curves have different degree you have to multiply each part of the equation with the degree of the corresponding curve.


In Topic: Why is infinite technically not a number.

05 June 2014 - 01:33 AM

@Mats1: As I said ealier, the result has been obtained using more complicated methods based on complex analysis and not using some kind of series manipulation. When you sum an infinite number of terms you have to be very careful to what you do. Most manipulations gives in fact wrong results. 


In Topic: Why is infinite technically not a number.

03 June 2014 - 06:57 AM

The series in the definition of the zeta function is convergent (in the usual sense) for complex numbers with real parts greater than 1. We may however give different definitions of this function with a greater domain and this analytic continuation is what we use to compute the zeta function of -1. 


In Topic: Can you quickly visualize how a quaternion would look on top of your head?

22 April 2014 - 07:36 AM

An axis can be defined by any non-zero vector, not just the unitary ones. You do not need to convert the imaginary part at all.. I can clearly take that imaginary part and normalize it if I need/want, but it isn't necessary to do so. If I look at my original example I can say that the axis is in the X>0,Y>0,Z<0 octant and that the greater component is along the z-axis. This component is indeed more than twice the x-component and more than three times the y-component. I would probably draw this axis using the vector (3,2,-7.1).


In Topic: Can you quickly visualize how a quaternion would look on top of your head?

22 April 2014 - 06:48 AM

If you take a general quaternion like 0.6 + 0.3*i + 0.2*j -  0.71*k you may first look at the real part and observe this should be equal to cos(alpha/2) for some angle alpha. 0.6 is greater than 0.5 which is the cosine of 60 degrees, but less then sqrt(2)/2 ~= 0.71 which is the cosine of 45 degrees. It should then be something in between, let's say 52.5 which is in the middle (the correct angle is 53.13 degrees) and thus we have a rotation of approximately 105 degrees around the axis defined by its imaginary part. This is basically how you can get a rough idea of what a quaternion does.


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