Determine vertical height of a geometry

[joeldiraviam]

Hi all,

I have attached a geometry. For this geometry I would like to determine vertical heights of two arcs(H2 and H3).

a) QR is a angle line with known length and angle
b) ST is an arc with known radius. The arc center lies on the whole geometry vertical centerline
c) RS is an arc with known radius. It is tangent to QR line and ST arc.

I need to find H1, H2 and H3. H1 is easy to find but the other two I find it difficult to get the equations. Can someone help me with the answers.

Also is there a way to find the RS arc centerpoint location based on the X and Y locations of the slant line QR. Any help will be greatly useful.

Thanks,
Joel

 

[jvicens]

Hi Joel,

Thank you for sharing your geometry and questions with us. I can provide some guidance on how to approach finding the vertical heights of the two arcs (H2 and H3) and the arc centerpoint location.

To find H2 and H3, we can use the Pythagorean theorem and trigonometric functions. Here are the steps:

1. First, we need to find the length of QR. Since we know the length and angle of QR, we can use the trigonometric function cosine (cos) to find the length of the adjacent side (which is QR). The formula is cos(angle) = adjacent/hypotenuse. In this case, cos(Q) = QR/H1. Solving for QR, we get QR = H1 * cos(Q).

2. Next, we can use the Pythagorean theorem to find the length of RS. The formula is c^2 = a^2 + b^2, where c is the hypotenuse (RS), and a and b are the other two sides (QR and ST). Substituting the values we know, we get RS^2 = (H1 * cos(Q))^2 + (H1 + r)^2, where r is the radius of ST. Solving for RS, we get RS = √[(H1 * cos(Q))^2 + (H1 + r)^2].

3. Now, to find H2 and H3, we can use the trigonometric function sine (sin). The formula is sin(angle) = opposite/hypotenuse. In this case, sin(H2 angle) = H1/(H1 + r). Solving for H2, we get H2 = (H1 + r) * sin(H2 angle). Similarly, sin(H3 angle) = (H1 + r)/RS. Solving for H3, we get H3 = RS * sin(H3 angle).

4. To find the arc centerpoint location, we can use the formula for the equation of a circle, which is (x - h)^2 + (y - k)^2 = r^2, where (h,k) is the centerpoint, and r is the radius. In this case, the centerpoint lies on the vertical centerline, so h = 0. We also know that the centerpoint is tangent to the QR line, so the distance between the center