Formula for Laying Out an ArcDate: 09/27/2002 at 12:09:16 From: Carl Propson Subject: Formula for laying out an arc using a known radius and distance from chord to the arc at 90 deg. Hello Dr. Math, I work as a layout carpenter and do not have the advantage of the newer transits. I am working on a building having a 317' radius with steel centers along it, and will not be able to swing this radius with a measuring tape. If I could figure the layout from a chord it would be a great help. There must be a formula for this. The radius is known, the distance from the center of the chord at 90 deg. to the arc is known, the distance from the center of the chord along the chord to a point is known - call it A. What would be the distance from A to the point where it intersects the arc at 90 deg. off the chord. Your help would be greatly appreciated. Thank you very much, Carl Date: 09/27/2002 at 15:34:54 From: Doctor Rick Subject: Re: Formula for laying out an arc using a known radius and distance from chord to the arc at 90 deg. Hi, Carl. I think I understand what you're saying, and it's not hard to work out using coordinate geometry. We can lay out our coordinate system so that the chord is along the x-axis with the center of the chord at the origin (0,0). If the radius is r and the perpendicular distance from the center of the chord to the arc is h, then the center of the circle is at (0, h-r), so the equation for the circle is x^2 + (y-(h-r))^2 = r^2 We can solve for y in terms of x: y = sqrt(r^2 - x^2) - r + h This is what you're looking for: x is the distance along the chord from the center of the chord to your point A, and y is the perpendicular distance from point A to the arc. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ Date: 09/29/2002 at 19:59:15 From: Carl Propson Subject: Formula for laying out an arc using a known radius and distance from chord to the arc at 90 deg. Hello Dr Rick, Thank you very much for all your help! Knowing that formula will make the job much easier. I do however have another question: is there a formula to determine the angle at the point where the line from point A running 90 deg. from the chord intersects the arc ? I must set the anchor bolts along the arc using this angle. Thanks again, Carl Date: 09/30/2002 at 08:26:55 From: Doctor Rick Subject: Re: Formula for laying out an arc using a known radius and distance from chord to the arc at 90 deg. Hi, Carl. Yes, the angle can be found using right-triangle trigonometry. We have a right triangle with one vertex at the center of the circle, C; another where the perpendicular line from the chord at A intersects the circle, which I'll call B; and the third where a line from the second vertex running parallel to the chord intersects the line through the circle center and the chord center - I'll call this D. The hypotenuse, CB, is the radius of the circle, r. The side opposite the center of the circle, BD, has length x. The sine of angle DCB is the ratio x/r. This angle is the same as the angle between the chord and the tangent to the circle at B (since CB is perpendicular to the tangent, and BD is perpendicular to the chord). Thus the angle you seek is arcsin(x/r). - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ Date: 10/26/2002 at 10:01:17 From: Carl Propson Subject: Thank you (Formula for laying out an arc using a known radius and distance from chord to the arc at 90 deg.) Thank you very much for your help; I am using these formlas daily in laying out the building. Thanks again, Carl |
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