# how to find coordinates of a point on intersection of arc and line

Here is an arc with known coordinates at starting & ending points.

The curve's starting & ending point coordinates are $A (0.19,0)$ and $B (0.1375,0.22)$ respectively.As curve is assumed to be a part of a circle whose center would be at $C (0,0.11)$ with radius $0.15$.

If a horizontal line (intersecting the curve) drawn at a distance $y=0.02$ ,what would be the possible method to find out the $x$ coordinate of that point of intersection?

• Can you show some work? Dec 23, 2014 at 9:59
• It seems a data is missing. There is an infinity of arcs that goes through $2$ specific points... You need the curvature, that is the center of the arc. Dec 23, 2014 at 10:18
• @ Julian Rachman- the curve's starting & ending point coordinates are A(0.19,0) & B(0.1375,0.22) respectively.As curve is assumed to be a part of a circle whose center would be at c(0,0.11) with radius 0.15. If a horizontal line(intersecting the curve) drawn at a distance y=0.02 ,what would be the possible method to find out the x coordinate of that point of intersection? Dec 23, 2014 at 10:20

## 1 Answer

The equation of the circle of center $C (0,0.11)$ and of radius $0.15$ is

$x^2+(y-0.11)^2=0.15^2$

The equation of the line is $y=0.02$

From there it is easy to find the intersection points.

And since you want only the point intersecting the arc, you have to see which point belongs to the arc.

• @Martigan- I have tried this method but when it is applied to verify the x coordinate of B, it gave wrong answer. Dec 23, 2014 at 10:45
• @user202851 The issue is that the radius you gave is clearly wrong. You can't have a center $C$ with a radius $0.15$ that goes through $A$. The distance $CA$ is greater than the radius... Dec 23, 2014 at 12:32
• @Martigan-I got your point that the center point assumed here is wrong. Can you suggest what it should be by observing the diagram? Dec 24, 2014 at 4:14
• @user202851 Since the distance between $A$ and $B$ is also greater than $0.15$ (easy to see because $y$ coordinates of $A$ and $B$ differ by $0.22$, which is greater than $0.15$), you have to assume that the radius is wrong. Since the distances between $C$ and $A$ and $C$ and $B$ are also different, you can't take any og them as the radius... You have no way of solving this. Dec 29, 2014 at 9:40
• Pro-tip: transform everything so the arc is on the unit circle. The math becomes easier. This will even let you find the intersection of an elliptical arc. Nov 18, 2015 at 20:19