Distance of a point (inside) circle with arbitrary direction
$begingroup$
For a circle with radius $r$ centered at point $A equiv (x_a, y_a)$,
How to calculate distance CM in, a given arbitrary direction
$d equiv (d_x, d_y) leftarrow |d|_2 = 1.0 $
for a point
$C equiv (C_x, C_y)$ inside the circle and a point on the Circle $M$
(see figure below)
geometry trigonometry euclidean-geometry
asked Dec 28 '18 at 7:14
DOOMDOOM
1848
$endgroup$
add a comment |
$begingroup$
For a circle with radius $r$ centered at point $A equiv (x_a, y_a)$,
How to calculate distance CM in, a given arbitrary direction
$d equiv (d_x, d_y) leftarrow |d|_2 = 1.0 $
for a point
$C equiv (C_x, C_y)$ inside the circle and a point on the Circle $M$
(see figure below)
geometry trigonometry euclidean-geometry
asked Dec 28 '18 at 7:14
DOOMDOOM
1848
$endgroup$
$begingroup$
Are you given point $C$ and some angle $theta$ and need to find $M$?
$endgroup$
– Daniel Mathias
Dec 28 '18 at 13:41
add a comment |
$begingroup$
For a circle with radius $r$ centered at point $A equiv (x_a, y_a)$,
How to calculate distance CM in, a given arbitrary direction
$d equiv (d_x, d_y) leftarrow |d|_2 = 1.0 $
for a point
$C equiv (C_x, C_y)$ inside the circle and a point on the Circle $M$
(see figure below)
geometry trigonometry euclidean-geometry
asked Dec 28 '18 at 7:14
DOOMDOOM
1848
$endgroup$
For a circle with radius $r$ centered at point $A equiv (x_a, y_a)$,
How to calculate distance CM in, a given arbitrary direction
$d equiv (d_x, d_y) leftarrow |d|_2 = 1.0 $
for a point
$C equiv (C_x, C_y)$ inside the circle and a point on the Circle $M$
(see figure below)
geometry trigonometry euclidean-geometry
geometry trigonometry euclidean-geometry
asked Dec 28 '18 at 7:14
DOOMDOOM
1848
asked Dec 28 '18 at 7:14
DOOMDOOM
1848
asked Dec 28 '18 at 7:14
DOOMDOOM
1848
asked Dec 28 '18 at 7:14
DOOMDOOM
1848
asked Dec 28 '18 at 7:14
DOOMDOOM
1848
1848
$begingroup$
Are you given point $C$ and some angle $theta$ and need to find $M$?
$endgroup$
– Daniel Mathias
Dec 28 '18 at 13:41
add a comment |
$begingroup$
Are you given point $C$ and some angle $theta$ and need to find $M$?
$endgroup$
– Daniel Mathias
Dec 28 '18 at 13:41
$begingroup$
Are you given point $C$ and some angle $theta$ and need to find $M$?
$endgroup$
– Daniel Mathias
Dec 28 '18 at 13:41
$begingroup$
Are you given point $C$ and some angle $theta$ and need to find $M$?
$endgroup$
– Daniel Mathias
Dec 28 '18 at 13:41
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The parametric equation of line $CM$ is
$$
(x,y)=(C_x+d_x,t,C_y+d_y,t).
$$
Substitute these coordinates into the equation of the circle
$$
(x-x_a)^2+(y-y_a)^2=r^2
$$
to get the values of $t$ giving the intersections. Of course you'll get two solutions for $t$: if you want only the intersection of the circle with the ray starting at $C$ with direction $d$, then you must keep only the positive solution.
answered Dec 28 '18 at 16:40
AretinoAretino
25.2k21445
$endgroup$
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The parametric equation of line $CM$ is
$$
(x,y)=(C_x+d_x,t,C_y+d_y,t).
$$
Substitute these coordinates into the equation of the circle
$$
(x-x_a)^2+(y-y_a)^2=r^2
$$
to get the values of $t$ giving the intersections. Of course you'll get two solutions for $t$: if you want only the intersection of the circle with the ray starting at $C$ with direction $d$, then you must keep only the positive solution.
answered Dec 28 '18 at 16:40
AretinoAretino
25.2k21445
$endgroup$
add a comment |
$begingroup$
The parametric equation of line $CM$ is
$$
(x,y)=(C_x+d_x,t,C_y+d_y,t).
$$
Substitute these coordinates into the equation of the circle
$$
(x-x_a)^2+(y-y_a)^2=r^2
$$
to get the values of $t$ giving the intersections. Of course you'll get two solutions for $t$: if you want only the intersection of the circle with the ray starting at $C$ with direction $d$, then you must keep only the positive solution.
answered Dec 28 '18 at 16:40
AretinoAretino
25.2k21445
$endgroup$
add a comment |
$begingroup$
The parametric equation of line $CM$ is
$$
(x,y)=(C_x+d_x,t,C_y+d_y,t).
$$
Substitute these coordinates into the equation of the circle
$$
(x-x_a)^2+(y-y_a)^2=r^2
$$
to get the values of $t$ giving the intersections. Of course you'll get two solutions for $t$: if you want only the intersection of the circle with the ray starting at $C$ with direction $d$, then you must keep only the positive solution.
answered Dec 28 '18 at 16:40
AretinoAretino
25.2k21445
$endgroup$
The parametric equation of line $CM$ is
$$
(x,y)=(C_x+d_x,t,C_y+d_y,t).
$$
Substitute these coordinates into the equation of the circle
$$
(x-x_a)^2+(y-y_a)^2=r^2
$$
to get the values of $t$ giving the intersections. Of course you'll get two solutions for $t$: if you want only the intersection of the circle with the ray starting at $C$ with direction $d$, then you must keep only the positive solution.
answered Dec 28 '18 at 16:40
AretinoAretino
25.2k21445
answered Dec 28 '18 at 16:40
AretinoAretino
25.2k21445
answered Dec 28 '18 at 16:40
AretinoAretino
25.2k21445
answered Dec 28 '18 at 16:40
AretinoAretino
25.2k21445
25.2k21445
add a comment |
add a comment |
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$begingroup$
Are you given point $C$ and some angle $theta$ and need to find $M$?
$endgroup$
– Daniel Mathias
Dec 28 '18 at 13:41