\point_to_line

Description 

Operator \point_to_line is a geometrical operator. It is useful to define points to draw a fold (mountain or valley) line. It allows to get vertices of a frequent origami fold, when a corner (or a general vertex) comes to a line. \point_to_line takes two points:

1. moving vertex,
2. pivot vertex

and also two edges:

1. the edge on which the point has to arrive,
2. the edge on which the intersection with fold line is found.

In fact, there are always two theoric solutions to set the moving point in the given edge (a segment can have up to two intersections with a circle). A last parameter can be given to specify if it is the first or the second intersection point met we want to consider. This parameter can be omitted, in this case the first intersection found around the circle is kept.

The return vertex is the intersection of the fold line and the given last edge.
This operator should only appear within a step block.

Figure: left: On this figure one can see both usages of \point_to_line showing both intersections of the circle made by i around o and the edge [da, bc].
right: Bringing i1 on edge [c, d] using d as pivot. Note that one have to specified the second intersection with [d, i2] anticlockwise.

Usage

  f = \point_to_line(i, o, [da, bc], [da, i]); % fig 9 left
  s = \point_to_line(i, o, [da, bc], [ab, b], second); % fig 9 left
  i2 = \point_to_line(i1, d, [c, d], [b, c], second);  % fig 9 right

Formal structure

  RETURN_VERTEX = \point_to_line(MOVING, PIVOT, LIMIT_EDGE, EDGE, WHICH);

Parameters

  MOVING
  Description    : Moving vertex.
  Type           : Symbol, vertex identifier.
  Allowed values : Any vertex identifier previously defined.
  PIVOT
  Description    : Pivot vertex. Movement circle radius is
                   [MOVING, PIVOT].
  Type           : Symbol, vertex identifier.
  Allowed values : Any vertex identifier previously defined.
  LIMIT_EDGE
  Description    : Indicates the movement segment limit. 
  Type           : Edge, couple of vertex identifiers
  Allowed values : Any couple of vertex identifiers already 
                   defined in a previous step.
  EDGE
  Description    : Indicates the edge on which intersection with 
                   fold line can be found. 
  Type           : Edge, couple of vertex identifiers
  Allowed values : Any couple of vertex identifiers already 
                   defined in a previous step.
  WHICH (optional)
  Description    : Flag indicates which point the operator returns.
  Type           : Symbol.
  Allowed values : One of [first, second]. 
  Default value  : first.

Return value

  RETURN_VERTEX
  Description    : New vertex intersection of fold line and EDGE.
  Type           : Symbol, vertex identifier.
  Allowed values : Any vertex identifier not already defined.

Conditions

• EDGE should have intersection with moving circle (PIVOT, [PIVOT, MOVING]).