Thinking about the issues in Dercuano drawings, I came up with some ideas that I think will be pretty interesting for interactive drawing. As usual, they are (sort of) algebraic and recursive.
The fundamental spatial element of the system is a curve, what you might call a path or a polyline. A curve is a continuous function from some interval of t points in time to (x, y) points in space, so it has direction and potentially variable velocity and duration as well as positions. (Maybe it also has a continuously varying pressure.) All curves start at time t=0, but they may have different end points.
The easiest way to create a new curve is to draw one with the mouse (which, in Some musings on applying Fitts’s Law to user interface design and data compression, I found had a bandwidth of about 6 bits per second) or a multitouch touchscreen (which I haven’t really tested yet but am hoping to find will have better bandwidth; it does have pressure). But you can also create curves that are straight lines or circles.
Each curve also has some kind of visual attributes, like width, color, transparency, blurriness, and some kind of noise texture, for each of stroke and fill. For right now, though, I’m less interested in those than about the purely positional ones.
Given a curve C, there are several unary operations that give new curves: C.reversed, which is the same positions in reverse order; C.normalized, which is the same positions with their time interval compressed or stretched to t ∈ [0, 1]; C.closed, which adds an instant straight-line jump back to the start; and C.constant, which makes the velocity along the curve constant, but doesn’t change its duration. If curves also carry pressure information, there’s also C.monoline, which sets the pressure to unity; C.brush, which sets the pressure to the reciprocal of the velocity; and C.invbrush, which sets the velocity to the reciprocal of the pressure. I want on-screen buttons of some kind to select these operations.
There are also operations to combine a curve, or set of curves, with a point: C + P translates the curve by the (x, y) coordinates in P, while P·C rotates and scales the curve (around its start point?) by the parameters in the point. I want interactive operations for invoking these with a point I specify interactively using the mouse or using pinch-zoom with two fingers. These two operations P₀·C + P₁ form a “frame of reference”.
Each point along a curve can be associated with a frame of reference in different ways, and another curve can be transformed by that frame of reference. First, there’s a translated frame of reference, where the rotate-and-scale part of the transformation is the identity. Second, there’s a translated-and-rotated frame of reference, where the rotate-and-scale part of the transformation scales by unity but rotates so that the tangent forward along the curve is always in the same direction, or in some arbitrary direction when the tangent doesn’t exist. Third, there are translated-and-rotated frames of reference where the scale is taken from the velocity or pressure of the curve.
The operation C₀.interpolate(C₁, C₂, N) produces a set of N curves. The first curve is C₁ translated and rotated to the beginning of C₀, and the last curve is C₂ translated and rotated to the end of C₀. The curves in between interpolate smoothly between them. Another similar operation does the same thing without the rotation. These operations are invocable in a direct-manipulation kind of way; initially C₁ and C₂ are the same, and N can be interactively adjusted up and down.
(Hmm, maybe this is the wrong formulation? Maybe I actually want to add C₁ and C₂ to the definition of C₀? In that case the objects become drawings rather than curves? Maybe I want to be able to interactively see the interpolated curves as I’m drawing C₀? Maybe drawing C₀ is, at least normally, a context-menu operation on C₁? What about adding noise to the interpolated curves?)
The idea is that a variety of visual repetition with variability, including things like grids, hatching on one side of a line, wood textures, starbursts, and tree branches, becomes straightforward and easy to express and to adapt, while capturing the real Kolmogorov complexity of the drawing in the data structure that is built up.