5 centered arch construction with a known Height & Width
I am trying to construct a precise 5 centered arch with a known Height & Width using a Pixel Processor contained in the software Substance Designer.
I found out this construction method from this image in the book “Practical Geometry for Builders and Architects” from 1921.
Set out the Span (JK= ½ span) and the rise KO.
With K as the center and radius KO describe the Arch Oa.
Then Ka = OK + the rise.
Describe a semicircle Ja. At K, the junction of the longer line JK with the shorter line Ka, produce KO to meet the semicircle at b.
Kb is the geometric mean or mean proportional between JK and KO.
From O set out Oe = Kb.
From O set out OV = span = 2JK. With J as center and radius Kb describe an arc bd
With V as center and radius Ve describe an arc ed cutting the arc bd at d**.**
From J set out Jr = ½ rise = OK/2.
Join Vd and produce to 3.
Join Vr. Join dr and produce to 1.
Then V, d, and r are the 3 centers for describing the 3 arches O_3; 3_1; and 1_J respectively with radii VO, d3 and r1.
The 2 other centers corresponding with r and d may be determined in a similar manner for describing the arcs which make up the remainder of the outline of the arch to the right of OK.
I followed all the instructions but I can not seem to obtain a perfect junction between the main arc O_3 and the other 2 on the right and 2 on the left (which instead are correctly merging). Here I link a picture with 3 cases showing the problem.
The arcs should connect perfectly. Can anyone please help me spot my mistake? Is it possible that this construction method is flawed? I am trying to find a 5 centered arch construction method that works. Many thanks
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