Curves in 3D - 3D Math Primer for Graphics and Game Development - page 651

see what p (t) looks like at the endpoints. We'll use the cubic case as our
example. At t = 0, we have

c 0 specifies the start
point

p (0) = c 0 + c 1 (0) + c 2 (0) 2 + c 3 (0) 3 = c 0 .

In other words, c 0 specifies the start point of the curve. Now let's see what
happens at the end of the curve at t = 1:

The endpoint is the sum
of the coe cients

p (1) = c 0 + c 1 (1) + c 2 (1) 2 + c 3 (1) 3 = c 0 + c 1 + c 2 + c 3 .

So the endpoint of the curve is given by the sum of the coe cients.

13.1.6 Velocities and Tangents

We can think of curves as being either static or dynamic. In the static
sense, a curve defines a shape. We operate in this mode of thinking when
we use a curve to describe the cross section of an airplane wing or a portion
of the letter “S” in the Times Roman font. In the dynamic sense, a curve
can be a trajectory or path of an object over time, with the parameter t as
“time” and the position function p (t) describing the position of a particle
at time t as it moves along the path.
If we consider only the static shape of the curve, then the timing of
the curve doesn't matter and our task is a bit easier. For example, when
defining a shape, it doesn't matter which endpoint is considered the “start”
and which is the ”end”; but if we are using the curve to define a path
traversed over time, then it matters very much where the path starts and
where it ends.
Using the dynamic mental framework and thinking about curves as
paths and not just shapes, some natural questions to ask are, “In what
direction is the particle moving at a given point in time?” “How fast is it
moving?” These questions can be answered if we create another function
v (t) that describes the instantaneous velocity of the particle at time t.
The phrase “instantaneous velocity” implies that the velocity changes
over time. So the next logical step is to ask, “How fast is the velocity
changing?” Thus it is also helpful to define an instantaneous acceleration
function a (t) that describes the rate at which the velocity of the particle is
changing at time t.
If you've had at least a semester of calculus, or if you read Chapter 11,
you should recognize that the velocity function v (t) is the first derivative
of the position function p (t) because velocity measures the rate of change
in position over time. Likewise, the acceleration function a (t) is the deriva-
tive of the velocity function v (t) because acceleration measures the rate of
change of velocity over time.