Variational geometric modeling with black box constraints and DAGs

• Extending geometric constraints with black box constraints which have no equation.
• Solving and optimizing without equation.
• This approach applies to all history-based or parametric modelers.

CAD modelers enable designers to construct complex 3D shapes with high-level B-Rep operators. This avoids the burden of low level geometric manipulations. However a gap still exists between the shape that the designers have in mind and the way they have to decompose it into a sequence of modeling steps. To bridge this gap, Variational Modeling enables designers to specify constraints the shape must respect. The constraints are converted into an explicit system of mathematical equations (potentially with some inequalities) which the modeler numerically solves. However, most of available programs are 2D sketchers, basically because in higher dimension some constraints may have complex mathematical expressions. This paper introduces a new approach to sketch constrained 3D shapes. The main idea is to replace explicit systems of mathematical equations with (mainly) Computer Graphics routines considered as Black Box Constraints. The obvious difficulty is that the arguments of all routines must have known numerical values. The paper shows how to solve this issue, i.e.,   how to solve and optimize without equations. The feasibility and promises of this approach are illustrated with the developed DECO (Deformation by Constraints) prototype.