![]() Because the sum of angles in a triangle is always 180 degrees, we now know that the angles φ and τ are: φ = τ = (180 - θ/2)/2 => φ = 90 - θ/4 If the full angle is cut in half - as shown with the blue angle η at figure 2 - we get an isosceles triangle (green) where the angles φ and τ are equal. The figures show a circle with a central angle describing an arc and we'll try to show that the yellow angles ε and σ are exactly one quarter of the cyan central angle θ. But why is the bulge 1/4 of the included angle and where does the tangent fit in? There are many ways to explain this. Now you have a bulge value for the arc segment in the polyline, and you can try out the formula above. Drop the endpoint somewhere, leave the polyline command and type this at the command line: Command: (setq ent (entget (entlast))) Start drawing a lightweight polyline, type "A" for arc, then "A" again for Angle and "120.0" for the included angle. So, a bulge of 0.57735 is describing an included angle of 2.09439 radians (which is 120.0 degrees, by the way). Simply use the built-in function ATAN to get an angle and multiply it by 4 in order to get the included angle: (* 4.0 (atan 0.57735)) In fact, once you have a bulge value, you can very quickly retrieve the included angle by inverting the above statement. That must be a clue of how to obtain the angle. Well, it also says that the bulge has something to do with the tangent of a quarter of the included angle of an arc. What does this mean and how can an arc be defined without even knowing the radius - or at least a chord length? It says that the only information given for arc segments in polylines are two vertices and a bulge. A bulge of 0 indicates a straight segment, and a bulge of 1 is a semicircle. A negative bulge value indicates that the arc goes clockwise from the selected vertex to the next vertex. So, what is a bulge for a circular arc and how is it defined? In AutoCAD's online help reference, it says about bulges for polylines: The bulge is the tangent of 1/4 of the included angle for the arc between the selected vertex and the next vertex in the polyline's vertex list. Both figures includes all of the attributes above, but for doing calculations with bulges, we'll mostly use the piece of pie that the arc cuts out of a circle, the circular sector. This line is drawn from the midpoint of an arc and perpendicular to its chord.Įxcept for the arc itself, an arc can describe two distinct geometric forms: Circular segment and circular sector. This line starts at the center and is perpendicular to the chord. ![]() ![]()
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