What does it mean by “the consecutive points marked on the curve appear at equal time intervals but not at...
What does it mean by
the consecutive points marked on the curve appear at equal time intervals but not at equal distances
?
parametric
asked Dec 26 at 14:52
user366312
473317
add a comment |
What does it mean by
the consecutive points marked on the curve appear at equal time intervals but not at equal distances
?
parametric
asked Dec 26 at 14:52
user366312
473317
add a comment |
What does it mean by
the consecutive points marked on the curve appear at equal time intervals but not at equal distances
?
parametric
asked Dec 26 at 14:52
user366312
473317
What does it mean by
the consecutive points marked on the curve appear at equal time intervals but not at equal distances
?
parametric
parametric
asked Dec 26 at 14:52
user366312
473317
asked Dec 26 at 14:52
user366312
473317
asked Dec 26 at 14:52
user366312
473317
asked Dec 26 at 14:52
user366312
473317
asked Dec 26 at 14:52
user366312
473317
473317
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
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."
answered Dec 26 at 20:09
David K
52.6k340115
add a comment |
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.
answered Dec 26 at 14:59
Ethan Bolker
41.3k547108
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
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."
answered Dec 26 at 20:09
David K
52.6k340115
add a comment |
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."
answered Dec 26 at 20:09
David K
52.6k340115
add a comment |
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."
answered Dec 26 at 20:09
David K
52.6k340115
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."
answered Dec 26 at 20:09
David K
52.6k340115
answered Dec 26 at 20:09
David K
52.6k340115
answered Dec 26 at 20:09
David K
52.6k340115
answered Dec 26 at 20:09
David K
52.6k340115
52.6k340115
add a comment |
add a comment |
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.
answered Dec 26 at 14:59
Ethan Bolker
41.3k547108
add a comment |
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.
answered Dec 26 at 14:59
Ethan Bolker
41.3k547108
add a comment |
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.
answered Dec 26 at 14:59
Ethan Bolker
41.3k547108
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.
answered Dec 26 at 14:59
Ethan Bolker
41.3k547108
answered Dec 26 at 14:59
Ethan Bolker
41.3k547108
answered Dec 26 at 14:59
Ethan Bolker
41.3k547108
answered Dec 26 at 14:59
Ethan Bolker
41.3k547108
41.3k547108
add a comment |
add a comment |
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