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Conic sections and quadric surfaces have been studied and applied over time because of their beautiful underlying geometry, aesthetically pleasing forms, and engineering functionality. One of the primary interests of this thesis is to gain a better understanding of conics and quadrics in a parametric Bernstein Bezier formulation; this formulation is advantageous to fully integrate these forms into computer aided geometric modeling. Applying this understanding, a boundary curve scheme for triangular interpolants is developed which has quadric precision. If the given point and normal data comes from a quadric, then the resulting boundary curves will lie on this quadric. Each boundary curve is a conic section, represented in rational quadratic Bezier form. Special care is taken to handle data which imply an inflection point. This boundary curve scheme is then integrated into a tangent continuous interpolation scheme, represented in terms of rational quartic Bezier triangular patches. Reflection lines and Gouraud shaded images are used as surface interrogation tools.