Branch angle

Basic angular value.

The local information about the branches is disregarded and the branches are seen as lines that connect the endpoints.

The endpoints are the origin of the tree, the nodes of the tree, and the terminations.

This simplification of the tree emphasizes the overall structure of the tree.

The angle is the change in direction from one branch to the next branch.

• Example: For a straight line with a node in the middle of the line, the planar angle is 0 -- not the 180º angle measured by placing a protractor at the node.

Change in direction using the line segments closest to the node (when branches are constructed of small line segments). Measured with the same method as the planar angle.

The local angle disregards the overall structure of the tree and concentrates on the information at the nodes. T

The local angle measurements can be prone to problems due to local errors. Part of the problem is due to the manner in which digital devices work. Coordinates are laid out in a checkerboard pattern. A point has only 8 neighbors, therefore, only 8 possible angles are available. The next ring of neighbors adds only 8 new possible angles. This means that, if the tracing is done with great care, the local angles are restricted to approximately 22 degree intervals. The planar angle does not have this problem since the endpoints are spaced far enough apart that the discrete nature of the measurement tools is not noticeable.

Alternate method to avoid the problems that can affect the local angle.

Since the simplest curves that can trace through 3D space are cubic curves, the branches are replaced with cubic splines. The splines smooth the segments. Tangents are taken at the ends of the cubic splines. The change in direction in the tangents is reported as the local spline angle.