            Construction idea Construction drawings Iso line drawings 2D construktion drawings Explaining of 3D congruence 3D construktion drawings Calculated 3D with "equal width"; Outerline design Calculated 3D for a real violin shape model Stradivarius iso lines measured by Möckel Guarneri del Gesu iso lines measured by Möckel Sacconi "belly" iso lines Sacconi "back" iso lines

Iso line drawings

How to check 3D congruence by comparing the shape of "elements-lines"

Regard a straight line from somewhere near the center of the shell to the outline to mark out the place of an "element".

The isoline drawings on the following pages have been completed with a number of such "elements". Equal numbers show where equal "elements" where found on both "upper" and "lower" parts of the arching.

To find out where equal "elements" are positioned one has to start marking out the place for one of them by drawing a straight line (marking out the place of the cross section to be checked). The location of an other one has to be searched for. The cross sectional shape of elements are equal when the distances between a number of intersection points with the isolines following on each other coincid with those already marked on the paper. Each isoline represent a very special height on the surface of the shell. The arcshape of an "element" that can be constructed is determined by the location of the intersection points of "the element line" with the "isolines".

This congruence can be checked in the following way:

1. Hold a piece of paper close to a drawn "element line" and mark the points of intersecting isolines on the edge of the paper.

2. Move the edge of the piece of paper around the isoline drawing in order to find the place where these marks coincid (intersect) with the isolines. When this happens an equal "element" has been found.

It is possible to find an almost countless number of such congruent pair of "element-lines" based upon the 3D geometry described on the previous pages. The computer calculated iso line layout for an arching shape with equal width for "upper" and "lower"parts This figure has no real violin isoline layout. It's only shown to illustrate the "element" congruence more obvious. The figure shows the iso line layout for a computer calculated arching shape with equal width for "upper and "lower" part. There is complete congruence of shape in each quarter (90 dgre sectors). The layout is completed with "elements" (straight lines) marking the place of 3 different cross sections in each quarter, which are equal in shape (congruent). The computer calculated iso-line layout for a normal violin arching with different width for "upper" and "lower" parts This is a correct violin iso-line model The figure shows computer calculated "iso lines" based upon the described 2D and 3D geometry, after the rotation of the main geometric layouts have taken place. These iso lines are comparable to the measured ones by Möckel. (Stradivarius and Guarnmeri del Gesu). The straight (1...4) equal numbered "elements" mark the exact places on the arching surface that are congruent. So the number 2 "elements" in the "upper part " are congruent to the number 2 in the "lower part" in spite of a different (wider-smaller) layout of the isolines.  The Stradivarius iso line layout measured by Otto Möckel The left figure shows the measured belly in its state of condition. In spite of the hand made shape it is easy to find congruent "elements" in and between both "upper" and "lower" part of the arching, here marked with equal line numbers. The right figure shows the measured back in its state of condition.  The Guarneri del Gesu iso line layout measured by Otto Möckel The left figure shows the measured belly in its state of condition. In spite of the hand made shape it is easy to find congruent "elements" in and between both "upper" and "lower" part of the arching, here marked with equal line numbers. The right figure shows the measured back in its state of condition. The Sacconi iso line layout for the "belly" as an assumption to createan "average" Stradivarius arching shape This figure shows Sacconis attemt to construct a 3D model by 2D geometric layout for a "belly" arching shape . It seems obvious that it is a 2D-construction since all intersection points are positioned on circular arcs intersecting straight lines. The figure is divided into 4 sectors marked by the 4 congruent shaped "elements" on the surface. Within the waist sector circumscribed by these 4 element lines there is complete congruence of shape not only comparing left and right but also up and down.. The dividing angle marksout the image line of the congruent picture. It is not possible to find any connection between the "upper" and "lower" sectors. The center of the "element" image is not in the center of the arching shape. No equal "elements" can be found covering the "upper" and "lower" parts of the shell. Only a limited sector in the "waist" has this quality. The Sacconi iso line layout of the "back arching" as an assumption to renderan "average" Stradivarius arching shape This figure shows Sacconis attemt to construct a 3D model by 2D geometric layouts for a "back" arching shape of an average Stradivarius violin. It seems obvious that it is a 2D-construction since all intersection points are positioned on circular arcs intersecting straight lines. The picture is divided into 4 sectors marked by the 4 congruent shaped "beams" on the surface. Within the waist sector circumscribed by these 4 line there is complete congruence of shape not only comparing left and right but also up and down.. The dividing angle marks out the image line of the congruent picture. It is not possible to find any connection between the "upper" and "lower" sectors. The center of the "element" image is not in the center of the arching shape. No equal "elements" can be found covering the "upper" and "lower" parts of the shell. Only a limited sector in the "waist" has this quality. 