\rabbit_ear

Description 

The geometrical operator \rabbit_ear has been designed to help users to make the special fold usually called "rabbit ear". The fold uses a triangle. The goal is to make three (usually valley) folds and one inverse fold (if the first three are valley folds the last is a mountain one). Each of the first three folds starts from a corner of the triangle and ends to a common intersection. This point is often the intersection of the three bisectors of each corner but can also be any given point inside the triangle.

As for other geometrical operators the goal is not to draw the result of the fold but to compute the new vertices needed to draw the fold lines. The goal of this operator is to provide a geometrical tool to compute how the last fold has to be done. In our case, there are two different syntaxes whether the common intersection of the three first folds is given or not. If this point is not given, we qualify the case as regular, which in fact is the most frequent case. In this situation, this common point is defined as the intersection of the three bisectors and it will be returned by the operator. In the second syntax (the goofy one), this point is given by user and already exists and it is not returned.

The last fold of the rabbit ear is what we are looking for; it starts from the common intersection of the three first folds and ends intersecting one of the three edge defining the initial triangle or another given edge. This latter point is the second point (or the only one) computed and returned by \rabbit_ear.

Inputs of this operator are:

• three vertices (defining the triangle, the rabbit ear fold reduces),
• the common intersection (optional),
• an edge, on which the searched vertex is expected (also optional).

So, as we can see one has to provide at least three vertex names as input of \rabbit_ear. Those points are not equivalent:

• the first one is the moving point of the rabbit ear fold. It is the point one grabs to make the fold.
• the second one can be called the destination point even if it gives only a direction rather than an exact position. This point indicates the direction the moving point takes during the fold.
• the third one is the last point defining the rabbit ear triangle.

Outputs of this operator can be:

• two new vertices (regular case),
• one new vertex (goofy case).

For the first case, the couple of returned vertices are given as extremities of an edge, but there are not an edge. It is just for convenient reasons as for the \line_to_line operator for instance (cf. §). Keep in mind that in this case, the output of this operator is not an edge but a couple of points. Nevertheless, this couple of points can also be seen as the last (mountain) fold described before, even if the physical edge has not been added to the edges data base. The order of vertices is important, the first one is the searched point located on one of the edge of the triangle and second one is the common center of the triangle.

Concerning the second syntax case, there is only one returned vertex because the common center has be given as a parameter.

This operator should only appear in a step block.

Figure: Example of a regular usage of the operator \rabbit_ear.
[bc, o1] = \rabbit_ear(b, c, a);
Point b is the moving point and it goes in the direction of the point c during the fold. As the common intersection has not been given it is computed as the intersection of the three bisectors. This point is o1 in the figure. The searched point bc is located on the edge [b, c] since the edge parameter has been omitted.
On the right the result of the fold is presented.

Figure: Example of a goofy usage of the operator \rabbit_ear.
bc = \rabbit_ear(c, ab, d, o, [b, c]);
Here, the common end of the three first valley fold is given. This point o is the center of the initial square (intersection of both diagonals). Moving point c goes in the direction of ab which is the middle point of the edge [a, b]. The searched point should be located on [b, c] which is not an edge defining the rabbit ear triangle.
On the right the result of the fold is presented.

Usage

  [bc, o1] = \rabbit_ear(b, c, a); % the simplest rabbit ear definition
  bc = \rabbit_ear(c, ab, d, o, [b, c]); % o is the square center

Formal structure

  [VERTEX, CENTER_O] = \rabbit_ear(MOVING, DEST, VERTEX_3, EDGE);
  [VERTEX, CENTER_O] = \rabbit_ear(MOVING, DEST, VERTEX_3);
  VERTEX = \rabbit_ear(MOVING, DEST, VERTEX_3, CENTER_I, EDGE);
  VERTEX = \rabbit_ear(MOVING, DEST, VERTEX_3, CENTER_I);

Parameters

  MOVING
  Description    : The corner of the triangle that moves during the
                   fold.
  Type           : Symbol, vertex identifier.
  Allowed values : Any vertex identifier previously defined.

  DEST
  Description    : One of both other points (than MOVING) of
                   the triangle to indicate the direction of
                   the rabbit ear.
  Type           : Symbol, vertex identifier.
  Allowed values : Any vertex identifier previously defined.

  VERTEX_3
  Description    : Third triangle corner vertex.
  Type           : Symbol, vertex identifier.
  Allowed values : Any vertex identifier previously defined.

  CENTER_I
  Description    : The vertex which is the intersection of the first
                   three folds of the rabbit ear.
  Type           : Symbol, vertex identifier.
  Allowed values : Any vertex identifier previously defined.

  EDGE (optional)
  Description    : The edge on which VERTEX should be found.
  Type           : Edge, couple of vertex identifier.
  Allowed values : Any couple of vertex identifiers 
                   previously defined.
  Default value  : [MOVING, DEST].

Return value

  VERTEX
  Description    : The searched vertex located on EDGE.
  Type           : Symbol, vertex identifier.
  Allowed values : Any vertex identifier not already defined.

  CENTER_O
  Description    : The intersection of three bisectors of the
                   rabbit ear triangle.
  Type           : Symbol, vertex identifier.
  Allowed values : Any vertex identifier not already defined.

Conditions

• MOVING, DEST and VERTEX_3 should not be equivalent (equivalence notion is explains in ) and not aligned,
• CENTER_I should be located inside the triangle.