Intersecting angle of point and line segment

calculusgeometrylinear algebratrigonometry

Given a line segment $L$ in $\mathbb R^2$ ($2D$-Space), represented by two points, and points $P_1$ and $P_2$ (also in $\mathbb R^2$) not on the segment, $L$ rotates about $P_1$, its full rotation forming an annulus (doughnut-like shape). Assuming that $P_2$ is inside of the annulus, I need to find the angle that $L$ needs to rotate about $P_1$ so that it intersects $P_2$ (such that $P_2$ lies on $L$ after its rotation).

The closest to an answer I've seen is this. It uses 3 lines in $\mathbb R^3$ (with one rotating about another to intersect with the last line), and I can imagine my problem being an analogue to this, the $\mathbb R^3$ space being projected onto $\mathbb R_2$ with two of the lines perpendicular to the plane of projection, though I struggle to understand how to execute this (and I believe there must be a simpler solution). Nor can I find a simple trigonometric answer.

I appreciate any and all help you can provide,
rbjacob

Best Answer

Just as @amd suggested:

Calculate $r= |P_2 - P_1|$.
Calculate $Q$, the intersection of the circle centered on $P_1$ of radius $r$, with $L$.
Calculate the angle $\theta$ from $P_1 Q$ to $P_1 P_2$.

Related Solutions

[Math] Distance between a circle and a line segment

Here is an example Python code that solves the optimization problem suggested by @DeepSea

import casadi as ca
opti = ca.Opti()

# Line segment
t = opti.variable()
line_seg = opti.bounded(0, t, 1)
p0 = (-4.0, 0.5)
p1 = (-2.0, 1.5)
x = (1 - t) * p0 + t * p1

# Circle
y = opti.variable(2)
circ = y[0]**2 + y[1]**2 == 1

# Optimization problem
dist_squared = (x[0] - y[0])**2 + (x[1] - y[1])**2
opti.minimize(dist_squared)
opti.subject_to(line_seg)
opti.subject_to(circ)

# Solve
opti.solver('ipopt')
sol = opti.solve()

# Result
print(f"Distance = {sol.value(ca.sqrt(dist_squared))}")
print(f"Point on line segment: {sol.value(x)}")
print(f"Point on circle: {sol.value(y)}")

Find the closest distance between a point and a segment defined by two points by coordinates

there is three area to take in count:

For $P1$ the distance is $|P1-B|$
For $P3$ the distance is $|P3-A|$

to summarise, it juste the distance between two point when the point P is not between the two perpendicular line passing trough A and B.

the hardest part is P2 :

since it between the two point, the distance to the segment is the distance to the point $C$ on the segment $AB$. We can know where the point C is located with a parameter $h$ (the doted line) because of how a segment is parameterised.

$$(A,B) = (hx_B + (1-h)x_A,hy_B + (1-h)y_A)$$

if C is on B then $h=1$ and $h=0$ if C is on A, in the exemple h might be equal to 0,75 or 0,8.

To find h we need to do :

$$h = \frac{<P_2-A, B-A>}{|B-A|^2}$$

this is a projection of the length of the vector $\overrightarrow{P_2A}$ on to $\overrightarrow{BA}$ we use a dot product for that, then devide it to the length of $\overrightarrow{BA}$ then normalise it ench the power of 2.

then to find the distance $|P2-C|$ we do :

$$|P_2 - (A +h*(B-A))|$$

to finish we can compute the 3 expression at the same time wich lead us to :

$$h = min(1,max(0,\frac{<P_2-A, B-A>}{|B-A|^2}))$$ $$d = |P - A - h*(B-A)|$$

Best Answer

Related Solutions

[Math] Distance between a circle and a line segment

Find the closest distance between a point and a segment defined by two points by coordinates

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