Obliquez

Obliquez is a theoretical construct in geometry used to study oblique coordinate systems and non-orthogonal projections. It denotes a class of coordinate frames together with a family of linear maps that emphasize oblique directions rather than perpendicular ones.

In an n-dimensional vector space, an obliquez frame is defined by an ordered basis B = (b1, ..., bn) consisting of linearly independent vectors. The basis induces an oblique coordinate system in which vectors are expressed as unique linear combinations of the basis vectors. The non-orthogonality is reflected in the Gram matrix G = [⟨bi, bj⟩], which is not constrained to be diagonal. The associated oblique projection refers to a linear map that projects onto a subspace along a complementary subspace in a non-orthogonal manner.

Properties include invertibility of the frame, since B is full rank, and a coordinate representation that remains unique despite non-orthogonality. The angles between basis vectors are generally not right angles, and the degree of obliquity influences the form of transformation matrices and numerical conditioning in computations.

Examples in two dimensions include a frame with b1 = (1, 0) and b2 = (1, 1), which yields a skewed oblique grid. The corresponding oblique projection onto the x-axis along the line y = x provides a simple demonstration of non-orthogonal projection behavior.

History and usage: The term obliquez is used in theoretical expositions to illustrate non-orthogonal coordinate systems and projections. It appears in discussions of graphics, mesh generation, and coordinate transformations that require shear and skew without enforcing orthogonality.

See also: Oblique projection; Non-orthogonal coordinates; Basis (linear algebra).