AGS - Curves

Today I'm going to start blogging a little about something that continually fascinates me: architectural geometry and surfaces (which I'll label as AGS from now on).  Whether it's surface, texture, contours, shape or volume, geometry is such a vital part of architecture that without it designing, as an artform, ceases to exist.  As a structural engineer, I understand that stiffness in structures doesn't exactly correspond to stiffness in materials, and that by extrapolating the correct structure based on overall shape - and not volume - one introduces deviatoric stresses throughout a whole integrated structure.  These principal stresses, primarily, are the foundation of all dynamically-responsive structures.  Therefore, the mastering of geometry is crucial to an artist/architect/engineer's overall education.

To begin with, I'm going to elucidate a little on something sexy: curves.  First of all, there are curve tangents, curvature of curves, inflection points of curves, and so on.  Curves are generally many things too, but for the purposes here, they are only thing explicitly: they are profiles used to generate surfaces.

The discussion of curves leads to the study of surfaces in a natural way.  Mathematically, this happens because of two analytical approaches to curves: (1) tangent planes and (2) surface normals.  A tangent plane is a manifold that facilitates the generalization of vectors (such as a surface normal) from affine spaces to general manifolds (surfaces).

  The Deep Blue Vector is a Surface Normal, the Transparent Blue Rectangle a Tangent Plane

A curve can also be more easily considered as a connected one-dimensional series of points.  As can be seen in a hyperbola, these point series can consist of different parts or the branches of a curve.  All of these one-dimensional curves are called planar curves, naturally, in contrast to spatial curves (such as helixes).

Next I will write about parametric curves, which are the curves used most for kinematic equations and structural dynamic analysis.   (Note: curves are both artistic and mathematical.  I will explain what I believe to be the overlap later in another post.)