.. status: ok



10.4 Points, Lines, and Curves
------------------------------

What you have seen so far is a high-level program using the graphics
facility. We now turn to the more basic notions of points, lines, and
curves in three-dimensional graphs. These facilities use small floats
(objects of type DoubleFloat) for data. Let us first give names to the
small float values 0 and 1.

The small float 0.


.. spadInput
::

	zero := 0.0@DFLOAT


.. spadMathAnswer
The small float 1.


.. spadInput
::

	one := 1.0@DFLOAT


.. spadMathAnswer
The @ sign means of the type. Thus zero is 0.0 of the type DoubleFloat.
You can also say 0.0::DFLOAT.

Points can have four small float components: x,y,z coordinates and an
optional color. A curve is simply a list of points connected by straight
line segments.

Create the point origin with color zero, that is, the lowest color on
the color map.


.. spadInput
::

	origin := point [zero,zero,zero,zero]


.. spadMathAnswer
Create the point unit with color zero.


.. spadInput
::

	unit := point [one,one,one,zero]


.. spadMathAnswer
Create the curve (well, here, a line) from origin to unit.


.. spadInput
::

	line := [origin, unit]


.. spadMathAnswer
We make this line segment into an arrow by adding an arrowhead. The
arrowhead extends to, say, p3 on the left, and to, say, p4 on the right.
To describe an arrow, you tell FriCAS to draw the two curves [p1,p2,p3]
and [p2,p4]. We also decide through experimentation on values for
arrowScale, the ratio of the size of the arrowhead to the stem of the
arrow, and arrowAngle, the angle between the arrowhead and the arrow.

Invoke your favorite editor and create an input file called
arrows.input. This input file first defines the values of

arrowAngle and arrowScale, then defines the function makeArrow (p1,p2)
to draw an arrow from point p1 to p2.


.. spadVerbatim

::


arrowAngle := %pi-%pi/10.0@DFLOAT               The angle of the arrowhead.

arrowScale := 0.2@DFLOAT                        The size of the arrowhead
                                                 relative to the stem.
 makeArrow(p1, p2) ==
   delta := p2 - p1                              The arrow.

  len := arrowScale * length delta              The length of the arrowhead.

  theta := atan(delta.1, delta.2)               The angle from the x-axis

  c1 := len*cos(theta + arrowAngle)             The x-coord of left endpoint

  s1 := len*sin(theta + arrowAngle)             The y-coord of left endpoint

  c2 := len*cos(theta - arrowAngle)             The x-coord of right endpoint

  s2 := len*sin(theta - arrowAngle)             The y-coord of right endpoint

  z  := p2.3*(1 - arrowScale)                   The z-coord of both endpoints

  p3 := point [p2.1 + c1, p2.2 + s1, z, p2.4]   The left endpoint of head

  p4 := point [p2.1 + c2, p2.2 + s2, z, p2.4]   The right endpoint of head

  [ [p1, p2, p3], [p2, p4] ]                    The arrow as a list of curves



Read the file and then create an arrow from the point origin to the
point unit.

Read the input file defining makeArrow.


.. spadInput
::

	)read arrows


.. spadMathAnswer
Construct the arrow (a list of two curves).


.. spadInput
::

	arrow := makeArrow(origin,unit)


.. spadMathAnswer
Create an empty object sp of type ThreeSpace.


.. spadInput
::

	sp := createThreeSpace()


.. spadMathAnswer
Add each curve of the arrow to the space sp.


.. spadInput
::

	for a in arrow repeat sp := curve(sp,a)


.. spadMathAnswer
Create a three-dimensional viewport containing that space.



vp := makeViewport3D(sp,"Arrow")





|picture|



Here is a better viewing angle.


.. spadInput
::

	rotate(vp,200,-60)


.. spadMathAnswer
|picture|






.. |picture| image:: ps/arrow.png
.. |picture| image:: ps/arrowr.png