Well, tgif doesn't really draw Bézier curves in general.
If there are 3 control points or less, what tgif draws is a
quadratic Bézier curve.
If there are n > 3 control points, tgif draws a
curve which is composed of n-2 pieces of curves
that, as a whole, is continuous, and has continuous
first derivatives everywhere (with approximations).
Since each piece has exactly 3 control points, each piece
is drawn as a quadratic Bézier curve.
So, I guess you can say that what tgif draws are
piecewise quadratic Bézier curves.
Below is basically where the n-2 pieces come from
for n > 3.
I will ilustrate this with an example for n = 5.
First, change the line type to straight, you should see n-1
line segments connecting the n original control points as shows in
Figure A.
Find the midpoints of all the middle n-3 line segments
and the 2 endpoints of the original curve, as shown in Figure B.
These are the endpoints for the quadratic Bézier curves.
This means that between any 2 consecutive points shown in Figure B,
tgif draws a quadratic Bézier curve. Therefore, there are a
total of n-2 quadratic Bézier curves. Please note that
between any of these 2 consecutive points (let's call them P1
and P2), there's a vertex (let's call it P3) which
is a control point of the original tgif spline object. From 3
control points, tgif already knows how to draw a quadratic Bézier curve
(see the top of this page). Tgif just uses the same algorithm to draw the
n-2 quadratic Bézier curves. Below is a more detailed
algorithm tgif uses for drawing (approximating)
a quadratic Bézier curve having
control points P1, P3, and P2.
The basic algorithm for drawing
a quadratic Bézier curve from control points P1,
P3, and P2 is depicted in Figure C below. The
quadratic Bézier curve is shown in red. Please note that the
curve is tangent to line segment P1-P3 at P1 and
it is also tangent to line segment P2-P3 at P2.
P4 is the midpoint of line segment P1-P3 and
P5 is the midpoint of line segment P2-P3.
Furthermore, P6 is the midpoint of line segment P4-P5.
The algorithm recursively draws two quadratic Bézier curves, one having
control points P1, P4, and P6 and one having
control points P2, P5, and P6.
Clearly, the curve and the first derivative of the curve are continuous
at point P6. Using this algorithm for Figure B, the 3
quadratic Bézier curve are shown in Figures D, E, and F.
Overlaying Figures D, E, and F together, the result is shown in
Figure G below.
