Calculating Partial Arcs

     Before a part can be programmed, the programmer must completely understand the part geometry. One of the more challenging shapes to program can be a partial arc radius.

     On a complete radius the feature begins at a specific point and travels 90 degrees to its ending point. The beginning and ending points are easy to calculate when you know the size of the radius.

     How do you calculate the beginning and ending points when the radius is only a partial arc? Let’s look at this part where the radius blends into a 15˚ angle (the black lines). We’ll review the steps to calculate the point where the radius ends and the angle begins. The first step is to sketch the theoretical sharp point between the front face and the angle (the red lines). For this example we are giving the diameter at this sharp point as 2.280”.

     There are 5 important points, identified on this print.

     When the coordinates for each of these points are known, it is much easier to program the radius feature. Let’s calculate each point.

P1 Arc start point in Z: This point is easy to identify. The arc starts on the front face and the front face is Z0. P1 = Z0.0

P2 Arc center point in Z: From the front face, move to the arc center -- this length is the radius, 0.5. P2 = Z-0.5

     These two points can be determined without mathematical calculations. The remaining points use the theoretical sharp point (2.280 diameter) and trigonometry to calculate the coordinates.

     Use trigonometry to calculate the length of the opposite side.
Opp = adj x tan 37.5
Opposite side = 0.38366

     Now we know the P3 arc start point is X is 2.880 – (2 x 0.38366) = 2.1127


P4 Arc ending point in X: Construct a triangle from the arc ending point as shown. The shaded angle is 15˚. The hypotenuse is the radius (0.5”) and the angle is 15˚. Use the cosine function to calculate the length of the adjacent side.

Adj = hyp x cosine 15
Adj = 0.4829629

     Use this value to calculate the unknown diameter value.
P3 + (2 x 0.4829629) = P4 P4 = X2.47786

P5 Arc ending point in Z: To find the last point, use the same 15˚ triangle to calculate the length of the opposite side.

Opp = hyp x sine 15
Opp = 0.129409

     Subtract this length from the known 0.500 length to determine the unknown length P5. 0.500 – 0.129409 = 0.3706 P5 = Z-0.3706

     With a little planning and a little trigonometry you can easily calculate the coordinates needed to program a partial arc. For more detailed images of the triangles and trigonometry, see our web site, www.cnc-training.com. Next month we’ll look to programming this basic radius feature.