Adding quadric fillets to quador lattice structures
AAdding quadric fillets to quador lattice structures
Fehmi Cirak ∗ , Malcolm Sabin Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
Abstract
Gupta et al. [1, 2] describe a very beautiful application of algebraic geometry to lattice structures composed of quadric of revolution(quador) implicit surfaces. However, the shapes created have concave edges where the stubs meet, and such edges can be stress-raisers which can cause significant problems with, for instance, fatigue under cyclic loading. This note describes a way in whichquadric fillets can be added to these models, thus relieving this problem while retaining their computational simplicity and e ffi ciency. Keywords: lattice structures, quadrics of revolution, quadors, algebraic geometryThroughout, we use an upper case letter to denote both asurface (e.g., a sphere S with the centre ( x c , y c , z c ) T and ra-dius r ) and the implicit function (i.e., S ( x , y , z ) = ( x − x c ) + ( y − y c ) + ( z − z c ) − r ) which is zero on that surface. No con-fusion should arise. Let S denote the quadratic function withthe zero set on the central sphere of a quador hub, increasingoutward from the centre. Figure 1: A hub (lattice joint) with two attached stubs (beams leading to otherjoints). This figure is diagrammatic only. Its purpose is to identify what surfacesthe various letters denote. The arrow on a surface indicates the gradient vectorof the expression whose zero set is the surface.
Let H and H denote the quadratic functions of two quadorstangent to the central sphere S . Then, these two functions havethe form H = S − G and H = S − G for linear functions G and G which are zero on the respectiveplanes of tangency, so that ∇ H = ∇ S and ∇ H = ∇ S .We can construct planes, for constants α > β > E = α F + + β F − and E = α F + − β F − ∗ Corresponding author with the functions F − = G − G and F + = G + G .Consider the quadrics whose equations are H − E = H along the curve of intersection with E ) and H − E = H along the curveof intersection with E ). These two will be the same quadric,providing a fillet between H and H if H − E = H − E or H − H = E − E but H − H = S − G − S + G = G − G = ( G + G )( G − G ) = F + F − and E − E = ( E + E )( E − E ) = (2 α F + )(2 β F − ) = αβ F + F − so we get a single fillet quadric if αβ = /
4. We can chooseeither α or β and then the other is fixed. The ratio betweenthe two controls the angles of the planes E and E either sideof F − . Increasing β slowly from zero increases the size of thefillet and its smallest radius of curvature, but this increases thelength of the stub.The above shows that there exists a fan of possible quadricsproviding a tangent continuous join between adjacent stubs.Each piece of surface has an exact implicit form, and an exactparametric form. The curves of tangency are all conics with ex-act parametric curves, exact implicit curves within their planes,and exact p-curves (i.e., trimming curves in parameter space)within both surfaces. Surfaces can be separated by the planeswhich contain the curves of tangency, and so all the importantproperties in [1] and [2] still apply. References [1] A. Gupta, G. Allen, J. Rossignac,
QUADOR: QUADric-Of-Revolutionbeams for lattices
CAD 102, 160–170 (2018)[2] A. Gupta, G. Allen, J. Rossignac,
Exact Representations and GeometricQueries for Lattice Structures with Quador Beams
CAD 115, 64–77 (2019)
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