cuspembedded

Cuspembedded is a term encountered in geometry and computer-aided design to describe the embedding of cusp-shaped features into higher-dimensional spaces in a way that preserves their singular structure.

Definition: It denotes an embedding of a cusp curve (for example, the plane cusp given by y^2 = x^3) into a larger ambient space such that the cusp remains recognizable in the image. The emphasis is on maintaining the local cusp geometry during the embedding, and on the image behaving coherently with respect to the ambient space.

Properties: A cuspembedded representation typically aims to balance identity of the cusp with the benefits of higher-dimensional placement, enabling analysis, rendering, or mesh generation that would be difficult in the original plane. It is discussed in contexts where singularities interact with embedding, such as in singularity theory and geometric modeling.

Applications: It is used in modeling cusp features in CAD and 3D graphics, geometry processing, and the study of how singularities behave under embedding or deformation.

Examples and relation: A familiar example is the standard plane cusp parameterized by t -> (t^2, t^3) viewed as a subset of R^3 by appending a coordinate, such as (t^2, t^3, 0). Related concepts include cusp singularities, embeddings, and immersions.

Terminology: The phrase cuspembedded is not widely standardized; in formal literature similar ideas appear under broader topics of singularities and embedded models.

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