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how to create Escher's knot in c4d

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Greg Sagehow to create Escher's knot in c4d
by on Dec 3, 2015 at 4:57:28 am

I'm creating a tunnel flythrough by sweeping a circle around a spline, tracking the camera along the spline, mapping a tesselation to it, etc.

I thought it might add something if I could have the spline follow the shape of Escher's Trefoil knot.

The wikipedia page for the knot:

Says it is defined by the following parametric equations:

x = sin t + 2sin 2t
y=cos t - 2cos 2t
z=-sin 3t

Really just want the most mathematically pure, evenly spaced, and most basic form of the knot, so want to create it by formula, and without variation, but struggling trying to figure out how to use the above equations to define a spline in C4d.

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Brian JonesRe: how to create Escher's knot in c4d
by on Dec 3, 2015 at 3:35:07 pm

just plug the equations into a Formula spline like this

The scale is pretty small with those numbers but you can either make it bigger by using multipliers in the math or just scale it up afterwards. It's also not a closed spline as parametric so it has to be made editable and closed there. If the length is one radian (6.283) it seems easiest to weld the two end points.

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Greg SageRe: how to create Escher's knot in c4d
by on Dec 6, 2015 at 2:42:40 am
Last Edited By Greg Sage on Dec 6, 2015 at 7:13:16 am

Thx. Never used a formula to generate a spline before. Didn't realize it was that easy.

Here's the result:

Working great with camera inside following spline except:

1) The only way I could figure out how to make it loop was to put keyframes on first frame, and one frame beyond last frame. Did this for camera, for camera target (null 7% ahead on spline), and light source (15% ahead on spline) It's not quite working, though. Even after I open up the f curves and make sure they're exactly on right values, and set to constant velocity, there's still a jump at the loop point. This is much worse for the other objects that were offset from 0 starting point since the program really doesn't seem to like looping from a negative value,or to one over 100%.

Is there a better way to handle this? One that is more flexible? If I just want to speed or slow the animation at this point, it's going to be a bit of a hassle as is.

2) I thought the rotations would be smooth, but they actually seem rather sudden. I see that the camera uses hpb rotation coordinates while following the spline, and that the bank value does not change. Is there perhaps a way to set some sort of formula for the bank to even it out... for instance so that if you swept the spline with a square, you could always have a certain side facing down? (at least from the perspective inside the knot)

3) Still trying to figure out how I can apply textures in After effects. Just not seeing that option anywhere, and I'm trying to wrap it with images that are dynamically created inside AE.

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