How should spatial motion tangents and shape tangents be interpreted?

Question

I'm writing a script that needs to interpret motion paths. The After Effects Scripting Guide specifies that each keyframe has a position, an in tangent and an out tangent.
My assumption was that these tangents indicate the position of the control points for a cubic bezier curve, relative to the keyframe position. So the curve from position p1 at keyframe 1 to position p1 at keyframe 2 is a cubic bezier determined by the points (p1,p1+out1,p2+in2,v2), where out1 is the out tangent of p1, and in2 is the in tangent of p2. Is this correct?
When I read about shapes, however, I got a bit confused. Shapes are also specified by vertices, and each vertex has an in-tangent and out-tangent. My initial assumption was that these also specify control points in a cubic bezier curve. But the guide shows an example of an oval:
As far as I know, there is no way to represent a circle or ellipse using cubic beziers. This got me thinking that perhaps my interpretation of the curves as cubic beziers is incorrect (I'd like to think that a circle in AE is really a circle, and not an approximation
Does anyone know how to interpret the tangents in both cases correctly?

Answer 1

Hi,
Bezier segments are polynomial curves, and as such, can't plot a circle (if X and Y are real polynomials then X^2+Y^2=1 would imply that both X and Y are constant, so a Bezier segment plotting exactly a portion of circle would plot at most one point of that circle...).
However they can approximate a circle (or ellipse) arc. This is what the example is talking about.
There is not just one way to decide when the approximation is good or not.
You can try to minimize the max radial distance from the approximated circle to the ideal one, or minimize the difference of areas, etc...
You'll get different data each time.
The most used convention for cubic Bézier and 90° arcs is "like this": start with a quarter of ideal circle, set the start and end points of the Bezier segment to coincide with the extremities of the arc, the tangents tangent to the arc, and adjust their common length so that the point at parameter t=0.5 coincides with the middle of the arc. Then (calculate) you find that the tangent length should be 4/3* (Math.sqrt(2)-1)  (0.552284741...) times the radius. This extends to 90° ellipse arcs as well since the image of a segment under affine transformation is the segment defined by the image of its control points. That's where the 55.23 comes from. If you want more info there are articles on the net about that (Cubic Bezier + circle).
There other approaches (other contraints)  that give closer approximations but for video it really doesnt matter, even 0.5 or 0.6 would work, noone will come and tell you that your circle is not a circle...
Your interpretation of motion path / shape path data is correct.
Xavier.