Auxiliary sphere method

The auxiliary sphere method is a method of descriptive geometry in order to determine the penetration curve (intersection curve) of two surfaces of revolution ( cylinder , cone , sphere , ...), whose axes of rotation intersect, point by point in a two- panel projection . An essential prerequisite is that the axes of rotation of the intersecting surfaces of revolution are parallel to one of the crack planes (floor plan or elevation). Because then the intersection circles of an auxiliary sphere, the center of which is the axis intersection, appear with the surfaces of revolution in a crack as lines.

If the axes do not intersect, but are horizontal or vertical, you should consider whether the auxiliary plane method can be used. The pendulum plane method offers a special alternative for the intersection of two cones or a cone with a cylinder .

Computational methods for determining points on an intersection curve are explained in the article Intersection curve .

Description of the process using an example

Auxiliary sphere method: cone-sphere intersection curve

Auxiliary sphere method: cone-sphere intersection curve, solution

A cone (axis ) and a cylinder (axis ) are given in plan and elevation (see picture). Find the penetration curve of the two surfaces. We choose spheres as auxiliary surfaces with the intersection of the axes as the center. Such balls with suitable radii intersect both the cone and the cylinder in circles as auxiliary curves. These circles are all perpendicular to the elevation board; that is, they appear as segments in the elevation. ${\displaystyle \Phi _{1}}$ ${\displaystyle a_{1}}$ ${\displaystyle \Phi _{2}}$ ${\displaystyle a_{2}}$ ${\displaystyle k=\Phi _{1}\cap \Phi _{2}}$ ${\displaystyle M=a_{1}\cap a_{2}}$

1. Choose a centered sphere that intersects both faces. ${\displaystyle \Psi }$ ${\displaystyle M}$
2. In the elevation, determine the intersection circles of the sphere with the cone and the sphere with the cylinder . We only use here . are lines because all circles are perpendicular to the elevation table. ${\displaystyle k_{1},l_{1}}$ ${\displaystyle \Phi _{1}}$ ${\displaystyle k_{2},l_{2}}$ ${\displaystyle \Phi _{2}}$ ${\displaystyle k_{2}}$ ${\displaystyle k''_{1},l''_{1},k''_{2},l''_{2}}$
3. ${\displaystyle k''_{1}\cap k''_{2}}$and provide the outline of max. four points of the penetration curve. It is . ${\displaystyle l''_{1}\cap k''_{2}}$ ${\displaystyle P,Q,R,S}$ ${\displaystyle P''=Q'',R''=S''}$
4. Draw and transfer to the floor plan using folders. lie on . ${\displaystyle k'_{1},l'_{1}}$ ${\displaystyle P,Q,R,S}$ ${\displaystyle P',Q',R',S'}$ ${\displaystyle k'_{1},l'_{1}}$
5. Repeat 1st to 4th n times.
6. Connect the points in the "correct" order with a curve.

literature

• Fucke, Kirch, Nickel: Descriptive Geometry. Fachbuch-Verlag, Leipzig, 1998, ISBN 3-446-00778-4
• Cornelie Leopold : Geometric Basics of Architecture , Verlag W. Kohlhammer, Stuttgart, 2005, ISBN 3-17-018489-X

Web links

• Descriptive geometry for architects (PDF; 1.5 MB). Script (Uni Darmstadt)