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.