Points, lines, and polygons

Section 3.5 Points, lines, and polygons

In this section, we will begin to introduce some simple graphical components, such as points, lines, vectors, and polygons.

Subsection 3.5.1 Points

Figure 3.5.1 shows a collection of points and various attributes describing them. The PreFigure source follows in Listing 3.5.2.

<diagram dimensions="(300, 300)" margins="5">
  <coordinates bbox="(0, 0,5, 5)">
    <point p="(1,4)" size="20" style="box"
	   fill="green" stroke="black" thickness="4"/>
    <point p="(2.5, 2.5)" size="15" style="diamond"
	   fill="blue" stroke="none"/>
    <point p="(3,1)" size="10"> <m>(a,(f(a))</m> </point>
    <point p="(1,1)" style="cross" thickness="3"/>
    <point p="(4,4)" style="plus" thickness="3"/>
  </coordinates>
</diagram>

Listing 3.5.2. The PreFigure source for Figure 3.5.1.

In the usual way, a point will have stroke and fill attributes. There are additional attributes that specify its appearance.

p: The @p gives the location at which the point will be centered.
style: This attribute controls the shape of the point. There are five options: @style="box", "diamond", "circle", "cross", "plus" with "circle" being the default.
size: This attribute specifies the size of the point in SVG units.

Notice that a label may be attached to a point by including the label’s text within the <point> element. In this case, attributes may be added to the point, such as @alignment and @offset, to control the positioning of the label. Behind the scenes, the label and point are placed within a <group> element, and the point’s handle transferred to the <group> for annotating.

Subsection 3.5.2 Lines

A few lines are shown in Figure 3.5.3 with the PreFigure source presented in Listing 3.5.4. Please note that a <vector> element can be used for including vectors in a diagram, as explained in Subsection 3.5.3.

<diagram dimensions="(300, 300)" margins="5">
  <coordinates bbox="(0,0,5,5)">
    <grid/>
    <line endpoints="((1,4), (4,4.5))" stroke="blue"
	  thickness="5" arrows="1"/>
    <line p1="(1,3)" p2="(4,2.5)" stroke="lightgray" thickness="20"/>
    <line p1="(1,3)" p2="(4,2.5)" endpoint-offsets="(10, -5)" thickness="5"/>

    <line endpoints="((1,1.5),(4,2))" stroke="green" infinite="yes"/>
    <definition>p1=(1,1)</definition>
    <definition>p2=(4.5,0.5)</definition>
    <line endpoints="(p1,p2)" endpoint-offsets="((10,0), (-10,0))"/>
    <label p="p1" clear-background="yes"><m>a</m></label>
    <label p="p2" clear-background="yes"><m>b</m></label>
  </coordinates>
</diagram>

Listing 3.5.4. The PreFigure source for Figure 3.5.3.

It is important to remember that lines, along with the other components introduced in this and later sections, support the graphical attributes explained in Section 3.1. In particular, any stroke attributes can be added to a line though fill attributes do not apply.

endpoints: The endpoints of a line may be specified together using the @endpoints attribute or separately using @p1 and @p2.
infinite: If @infinite="yes", then the line is drawn as if it were infinite in extent.
arrows: If @arrows="1", then an arrow is drawn at the second endpoint. If @arrows="2", then an arrow is drawn at both endpoints.
reverse: Setting @reverse="yes" changes the direction of the line when adding arrows.
endpoint-offsets: Sometimes you may want the line to begin or end near, but not exactly, at the given endpoints. The bottom line in Figure 3.5.3 is an example in that the line is connecting two labels placed at two endpoints. We move the endpoints slightly so that the line does not interfere with the label. In this case, the endpoint offsets are a pair of vectors, each of which moves an endpoint by the given vector in SVG units.

Another line in the figure shows a second use of offsets. In this case, the offsets are a single pair of SVG units. The direction of the line is preserved but each endpoint is moved along the line by the given offset. This could be helpful when drawing arrows since the arrow head will extend slightly beyond the endpoint of the line.

Subsection 3.5.3 Vectors

In some sense, <vector>s are a redundant element since we can always draw lines and attach arrows. However, a goal of PreFigure is to facilitate authors thinking mathematically about the exposition so <vector> should be used when including a vector in a diagram. Vectors are also drawn with a default thickness that differentiates them from lines drawn with their default attributes.

<diagram dimensions="(300, 300)" margins="5">
  <coordinates bbox="(-1,-1,5,5)">
    <grid-axes decorations="no"/>
    <vector v="(3,4)"/>
    <vector v="(1,3/2)" tail="(2,1)" scale="2" stroke="blue"/>
    <vector v="(1,3)" tail="(3.5,1)" head-location="0.8"/>
  </coordinates>
</diagram>

Listing 3.5.6. The PreFigure source for Figure 3.5.5.

In addition to stroke attributes, a vector supports the following attributes.

v: The attribute @v is the 2-dimensional mathematical vector.
tail: The attribute @tail represents the tail of the vector.
scale: The value of @scale acts as scalar multiplication on @v.
head-location: Usually, the head of a vector is placed at the tip of the vector. There may be occasions, however, when you want to place the head somewhere else along the shaft. Specifying @head-location="0.8" places the arrow head at a location 80% of the way from the tail to the tip.

Subsection 3.5.4 Polygons

Polygons are paths drawn by connecting a set of points by line segments. The effect is different from drawing a set of line segments, however, since the ends of the line segments will be joined to form a whole. For example, the shape on the left of Figure 3.5.7 is drawn with two <line>s while that on the right is a <polygon>.

Figure 3.5.7. Two line segments on the left and a polygon on the right.

<polygon>s are quite versatile and should more properly be thought of as a polygonal path having both stroke and fill attributes. Figure 3.5.8 demonstrates some possibilities with the PreFigure soure given in Listing 3.5.9.

<diagram dimensions="(300,300)" margins="5">
  <coordinates bbox="(-5,-5,5,5)">
    <grid/>
    <polygon points="((-2,-1),(-4,-1),(-4,2))" closed="yes"
	     fill="blue"/>
    <polygon points="(2,1)+2*(cos(k*pi/4),sin(k*pi/4))"
	     parameter="k=0..7" closed="yes"
	     stroke="red" thickness="4"/>
    <polygon points="((-4,4),(-1,4),(-1,-3),(3,-3))"
	     corner-radius="10" arrows="1" thickness="5"
	     stroke="green"/>
  </coordinates>
</diagram>

Listing 3.5.9. The PreFigure source for Figure 3.5.8.

A <polygon> requires a set of points given by the @points attributes. While you may simply list the points, PreFigure also allows you to provide a rule for generating the list of points.

points: The value of the @points attribute may be a list of points or a single point containing a named parameter.
parameter: The @parameter attribute gives a range of values for generating the list of points. The format is @parameter="parameter_name=start..stop".
closed: By default, polygons are not closed, but @closed="yes" will close a polygon.
arrows: To place arrows at the end of a polygonal path, use @arrows="1" to add an arrow at the end and @arrows="2" to also add an arrow at the beginning.
reverse: Setting @reverse="yes" changes the direction of the polygonal path when attaching arrows.
corner-radius: A polygonal path can be smoothed out using the @corner-radius attribute. The value of this attribute should be a positive number of SVG units.

Subsection 3.5.5 Splines

PreFigure can create cubic splines from a sequence of points. Figure 3.5.10 illustrates a number of possibilities.

Figure 3.5.10. A number of cubic splines

<diagram dimensions="(300,300)" margins="5">
  <coordinates bbox="(-4,-4,4,4)">
    <definition>f(k)=3*(cos(pi*k/2), sin(pi*k/2))</definition>
    
    <grid-axes xlabel="t" ylabel="y"/>
    <spline points="(0,2,1,2,4)" stroke="red"/>
    <spline points="((1,0),(2,0),(2,1),(1,2),(-1,2),(-2,0),(-1,-2),(1,0))"
            t-values="(0,2,3,3.5,4,5,7,9)"
            closed="yes" stroke="blue"/>
    <spline points="f(k)" parameter="k=0..4" closed="yes" stroke="orange"/>
    <circle center="(0,0)" radius="3" stroke="green"/>
  </coordinates>
</diagram>

Listing 3.5.11. The PreFigure source for Figure 3.5.10.

The <spline> defined in line 6 is a simple one-dimensional spline formed from a sequence of \(y\)-values. By default, PreFigure assumes that the corresponding \(t\) values are 0, 1, 2, 3, 4. This can be modified using the @t-values attribute.

The second <spline> defined on lines 7-9 is given by a sequence of 2-dimensional points. An accompanying sequence of @t-values is also defined. The attribute @closed="yes" asks that the spline be closed and a periodic boundary condition imposed. In this case, it is important that the last point agree with the first point.

If the spline is not to be closed, then boundary conditions can be specified using the @bc attribute. The default value is not-a-knot, but other possibilities can be found in the scipy.interpolar.CubicSpline documentation.

Since a <spline> is constructed from a sequence of points, a <spline> shares many attributes with a <polygon>. For instance, the spline defined in line 10 shows how the sequence of points can be defined using a @parameter. In addition, a <spline> can have a @stroke and @fill attribute, and @arrows can be used to add arrows.

Subsection 3.5.6 Triangles

For convenience, PreFigure offers a <triangle> as illustrated in Figure 3.5.12.

<diagram dimensions="(300,300)" margins="5">
  <coordinates bbox="(0,0,10,10)">
    <grid-axes decorations="no"/>
    <triangle vertices="((1,1),(3,2),(1,5))" labels="A,B,C"
	      show-vertices="yes" point-fill="orange" />
    <definition>vertices=((4,4), (8,1),(7,8))</definition>
    <triangle vertices="vertices" labels="D,E,F"
	      angle-markers="yes"/>

  </coordinates>
</diagram>

Listing 3.5.13. The PreFigure source for Figure 3.5.12.

A <triangle> has the same attributes as a <polygon>, except that the vertices are listed with the @vertices attribute and the attribute @closed is always set to "yes" for a <triangle>. There are some additional attributes of note.

labels: You may provide labels for each of the vertices by setting this attribute to a comma-separated list of three labels. Each label will be wrapped in an <m> tag and processed by MathJax.
angle-markers: Set this to "yes" to place an <angle-marker> in each of the interior angles.
show-vertices: This attribute will place a <point> on each vertex if set to "yes".
point-fill: Use this to specify the fill of each <point> if @show-vertices="yes".

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