What does it mean by "the consecutive points marked on the curve appear at equal time intervals but not at equal distances"?

Question

What does it mean by

the consecutive points marked on the curve appear at equal time intervals but not at equal distances

?

Answer 1

Look at Figure $2.$ See the points labeled $t=-2,$ $t=-1,$ $t=0,$ $t=1,$ and so forth. Notice that each of these values of $t$ is exactly $1$ unit greater than the one before it. So the time intervals (the increases in $t$) are equal.

Now look at where each of these points is plotted on the curve. Look at how long the segment of the curve between $t=1$ and $t=2$ is. Look at how long the segment of the curve between $t=3$ and $t=4$ is. Do those segments have equal lengths? That is what the book means by "not at equal distances."

Answer 2

The picture shows a curve described by a parametric equation with points marked corresponding to equally spaced values of the parameter, thought of as representing time taken to traverse the curve. Since the speed along the curve can vary with time, the successive points on the curve are not the same distance apart along the curve.

For example, consider the unit circle parameterized by $$ f(t) = (\cos(t^2), \sin(t^2)). $$ If you mark the points corresponding to integral values of $t$ they will be spaced further and further apart along the circle as $t$ increases.