Examples with source

Chapter 7 Examples with source

This chapter presents a series of PreFigure diagrams along with their source. These examples may provide a starting point for authors to explore in the PreFigure Playground.

davidaustinm.github.io/prefigure

Two points on a line with the horizontal and vertical changes indicated. — Figure 7.0.1. The definition of slope

Source.

Listing 7.0.2. The source for Figure 7.0.1

<diagram dimensions="(300,300)" margins="5">
  <definition>x1=1</definition>
  <definition>x2=4</definition>
  <definition>y1=1</definition>
  <definition>y2=5</definition>
  <definition>p=(x1,y1)</definition>
  <definition>q=(x2,y2)</definition>
  <definition>r=(x2,y1)</definition>
  <coordinates bbox="(-1,-1,6,6)">
    <axes decorations="no" xlabel="x" ylabel="y"/>
    <group outline="none">
      <polygon points="(p,r,q)" stroke="red"/>
      <angle-marker points="(r,p,q)" stroke="black">
        <m>\theta</m>
      </angle-marker>
      <line endpoints="(p, q)" stroke="blue" infinite="yes"/>
    </group>
    <label anchor="((x1+x2)/2,y1)" alignment="south">
      <m>\Delta x</m>
    </label>
    <label anchor="(x2,(y1+y2)/2)" alignment="east">
      <m>\Delta y</m>
    </label>
  </coordinates>
</diagram>

Two lines intersect in a point with the angle between the lines indicated. — Figure 7.0.3. Intersections of lines

Source.

Listing 7.0.4. The source for Figure 7.0.3

<diagram dimensions="(300,300)" margins="5">
  <coordinates bbox="(0,0,10,10)">
    <grid-axes decorations="no"/>
    <definition>p1=(1,2)</definition>
    <definition>p2=(8,8)</definition>
    <definition>L1=(p1,p2)</definition>
    <definition>q1=(1,7)</definition>
    <definition>q2=(8,4)</definition>
    <definition>L2=(q1,q2)</definition>
    <line endpoints="L1" stroke="blue"
      infinite="yes"/>
    <line endpoints="L2" stroke="blue"
      infinite="yes"/>
    <definition>p=intersect((L1,L2))</definition>
    <angle-marker points="(q2,p,p2)"
      clear-background="yes">
      <m>\theta</m>
    </angle-marker>
    <point p="p"/>
  </coordinates>
</diagram>

Two triangles, one of which is a right triangle. — Figure 7.0.5. Sample triangles

Source.

Listing 7.0.6. The source for Figure 7.0.5

<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>

The graph of a function with two vertical and one horizontal asymptotes. — Figure 7.0.7. Graphing functions with asymptotes

Source.

Listing 7.0.8. The source for Figure 7.0.7

<diagram dimensions="(300,300)" margins="0">
  <definition>f(x)=(x^2-1)/(x^2-4)</definition>
  <coordinates bbox="(-5,-5,5,5)">
    <grid-axes/>
    <graph function="f"/>
    <line endpoints="((-2,-4),(-2,4))" infinite="yes" stroke="red"/>
    <line endpoints="((2,4),(2,-4))" infinite="yes" stroke="red"/>
    <line endpoints="((4,1),(-4,1))" infinite="yes" stroke="green"/>
  </coordinates>
</diagram>

A point on the unit circle with its reflection in the horizontal axis. — Figure 7.0.9. A trigonometric identity

Source.

Listing 7.0.10. The source for Figure 7.0.9

<diagram dimensions="(300,300)" margins="5">
  <definition>theta=radians(80)</definition>
  <definition>p=(cos(theta),sin(theta))</definition>
  <definition>q=(cos(theta),-sin(theta))</definition>
  <definition>origin=(0,0)</definition>
  <definition>one=(1,0)</definition>
  <coordinates bbox="(-1.5,-1.5,1.5,1.5)">
    <grid-axes decorations="no"/>
    <polygon points="(one,origin,p)" stroke="red"/>
    <polygon points="(one,origin,q)" stroke="red"/>
    <circle center="(0,0)" radius="1" stroke="blue"/>
    <point p="p">
      <m>(\cos(\theta),\sin(\theta))</m>
    </point>
    <point p="q" alignment="se">
      <m>(\cos(\theta),-\sin(\theta))</m>
    </point>
    <angle-marker points="(one,origin,p)" stroke="black">
      <m>\theta</m>
    </angle-marker>
    <angle-marker points="(q,origin,one)" stroke="black">
      <m>-\theta</m>
    </angle-marker>
  </coordinates>
</diagram>

A right triangle with squares built on each of the sides. — Figure 7.0.11. The Pythagorean theorem

Source.

Listing 7.0.12. The source for Figure 7.0.11

<diagram dimensions="(250,270)" margins="5">
  <definition>a=3</definition>
  <definition>b=4</definition>
  <definition>c=sqrt(a^2+b^2)</definition>
  <definition>phi=acos(a/c)</definition>
  <definition>d=a*cos(phi)</definition>
  <definition>p=a*(cos(phi), sin(phi))</definition>
  <coordinates bbox="(-2.5,-5,7.5,6)" aspect-ratio="1">
    <rectangle lower-left="(0,-c)" dimensions="(d,c)"
               fill="skyblue"/>
    <rectangle lower-left="(d,-c)" dimensions="(c-d,c)"
               fill="springgreen"/>
    <transform>
      <rotate by="phi-pi/2" degrees="no"/>
      <rectangle lower-left="(-a,0)" dimensions="(a,a)"
                 fill="skyblue" stroke="black"/>
      <rectangle lower-left="(0,a)" dimensions="(b,b)"
                 fill="springgreen" stroke="black"/>
    </transform>
    <rectangle lower-left="(0,-c)" dimensions="(c,c)"
               stroke="black"/>
    <line endpoints="(p,(d,-c))" dash="8 8" stroke="black"/>
    <triangle at="triangle" vertices="((0,0),(c,0),p)"
              stroke="black"/>
  </coordinates>
</diagram>

The eight roots of unity plotted on the unit circle. — Figure 7.0.13. Eighth roots of unity

Source.

Listing 7.0.14. The source for Figure 7.0.13

<diagram dimensions="(300,300)" margins="5">
  <definition>
    alignments=['ne','ne','ne','nw','nw','sw','se','se']
  </definition>
  <definition substitution="no">
    labels=['1','\omega','i','\omega^3','-1','\omega^5','-i','\omega^7']
  </definition>
  <definition>f(t)=(cos(pi*t/4),sin(pi*t/4))</definition>
  <coordinates bbox="[-1.4,-1.4,1.4,1.4]">
    <grid at="grid" />
    <axes at="axes" labels="no" />
    <circle at="unit-circle" center="(0,0)" radius="1" stroke="blue"/>
    <repeat parameter="k=0..7">
      <point at="point" p="f(k)" alignment="alignments[k]">
        <m>${labels[k]}</m>
      </point>
    </repeat>
  </coordinates>

</diagram>

The graphs of a function, its derivative, and its second derivative. — Figure 7.0.15. Graphing derivativees

Source.

Listing 7.0.16. The source for Figure 7.0.15

<diagram dimensions="(300,300)" margins="5">
  <definition> f(x)=0.1*(x^4-10*x^2)</definition>
  <definition> fp(x)=deriv(f, x)</definition>
  <definition> fpp(x)=deriv(fp,x)</definition>
  <coordinates bbox="[-4,-4,4,4]">

    <grid-axes xlabel="x" ylabel="y"/>
    <graph function="f"/>
    <label p="(3.3,1)" alignment="center" clear-background="yes">
      <m>f</m>
    </label>
    
    <graph function="fp" stroke="red"/>
    <label p="(2.5,1)" alignment="center" clear-background="yes">
      <m>f'</m>
    </label>
    
    <graph function="fpp" stroke="green"/>
    <label p="(1.6,1)" alignment="center" clear-background="yes">
      <m>f''</m>
    </label>
    
  </coordinates>
</diagram>

The graph of a function, a tangent line, and the intersection of the tangent line with the horizontal axis. — Figure 7.0.17. Newton’s method

Source.

Listing 7.0.18. The source for Figure 7.0.17

<diagram dimensions="(300,300)" margins="5">
  <definition>f(x)=7-(x+1)^2/2</definition>
  <definition>x0 = 1</definition>
  <coordinates bbox="(-8,-8,8,8)">
    <grid-axes xlabel="x" ylabel="y"/>
    <graph function="f"/>
    <tangent-line function="f" point="x0" name="L"/>
    <point p="(x0,f(x0))" alignment="se"/>
    <definition>x1 = intersect(L,-1)</definition>

    <repeat parameter="x in (x0, x1)">
      <line endpoints="((x,0),(x,0))"
            endpoint-offsets="((0,-3),(0,3))"
            tactile-endpoint-offsets="((0,-18),(0,0))"
            stroke="black"/>
    </repeat>
    <label anchor="(x0,0)" alignment="north">
      <m>x_0</m>
    </label>
    <label anchor="(x1,0)" alignment="ne">
      <m>x_1</m>
    </label>
  </coordinates>
</diagram>

The graphs of the sine and cosine functions with the area between them indicated. — Figure 7.0.19. An intergral measuring area

Source.

Listing 7.0.20. The source for Figure 7.0.19

<diagram dimensions="(300,300)" margins="5">
  <definition>f(x)=sin(x)</definition>
  <definition>g(x)=cos(x)</definition>
  <definition>a=intersect((f,g),1)</definition>
  <definition>b=intersect((f,g),4)</definition>
  <coordinates bbox="(-1,-1.5,5,1.5)">
    <grid-axes decorations="no"/>
    <area-between-curves functions="(f,g)" domain="(a,b)" fill="lightgray"/>
    <graph function="f"/>
    <graph function="g"/>
    <line endpoints="((a,0),(a,0))" endpoint-offsets="((0,-3),(0,3))"/>
    <label anchor="(a,0)" alignment="south" offset="(0,-7)">
      <m>a</m>
    </label>
    <line endpoints="((b,0),(b,0))" endpoint-offsets="((0,-3),(0,3))"/>
    <label anchor="(b,0)" alignment="south" offset="(0,-4)">
      <m>b</m>
    </label>

  </coordinates>
</diagram>

The slope field of an ordinary differential equation and solutions corresponding to several initial values. — Figure 7.0.21. Solutions to a differential equation

Source.

Listing 7.0.22. The source for Figure 7.0.21

<diagram dimensions="(300,300)" margins="5">
  <definition>f(t,y) = t-y</definition>
  <coordinates bbox="[-4,-4,4,4]">
    <grid at="grid"/>
    <axes at="axes" xlabel="t" ylabel="y"/>
    <slope-field at="slope-field" function="f" outline="yes"/>
    <group at="solutions" outline="tactile">
      <repeat parameter="k=-4..4">
        <plot-de-solution at="solution" function="f" t0="0" y0="k" t1="4" stroke="orange"/>
        <point at="initial-value" p="(0,k)" size="4" fill="orange"/>
      </repeat>
    </group>
  </coordinates>
</diagram>

The solution to a system of differential equations. — Figure 7.0.23. A system of differential equations

Source.

Listing 7.0.24. The source for Figure 7.0.23

<diagram dimensions="(300,300)" margins="5">
  <definition>f(t,y) = (y[1], -pi*y[0]-0.3*y[1])</definition>
  <coordinates bbox="[-1,-3,6,3]">
    <grid-axes xlabel="t"/>
    <de-solve function="f" t0="0" t1="bbox[2]" y0="(0,2)"
              name="oscillator" N="200"/>
    <plot-de-solution at="x" solution="oscillator" axes="(t,y0)"/>
    <plot-de-solution at="xprime" solution="oscillator"
                      axes="(t,y1)" stroke="red" tactile-dash="9 9"/>
    <legend at="legend" anchor="(bbox[2], bbox[3])" alignment="sw"
            scale="0.9" opacity="0.5">
      <item ref="x"><m>x(t)</m></item>
      <item ref="xprime"><m>x'(t)</m></item>
    </legend>
  </coordinates>
</diagram>

Two two-dimensional vectors v1 and v2 and a linear combination of them. — Figure 7.0.25. The linear combination of vectors

Source.

Listing 7.0.26. The source for Figure 7.0.25

<diagram dimensions="(300,300)" margins="5">
  <definition>v1 = (2,1)</definition>
  <definition>v2 = (-1,1)</definition>
  <definition>a=3</definition>
  <definition>b=2</definition>
  <definition>w=a*v1+b*v2</definition>
  <coordinates bbox="(-3,-3,8,8)">
    <axes decorations="no"/>
    <grid basis="(v1,v2)" stroke="lightgray"/>
    <group outline="tactile">
      <vector v="v1" stroke="blue"/>
      <vector v="v2" stroke="blue"/>
      <vector v="w" stroke="red"/>
    </group>
    <label anchor="v1" alignment="e" clear-background="yes" offset="(5,0)">
      <m>{\mathbf v}_1</m>
    </label>
    <label anchor="v2" alignment="n" clear-background="yes" offset="(0,5)">
      <m>{\mathbf v}_2</m>
    </label>
    <label anchor="w" alignment="n" clear-background="yes" offset="(0,5)">
      <m>${a}{\mathbf v}_1+${b}{\mathbf v}_2</m>
    </label>
  </coordinates>
</diagram>

The orthogonal projection of a vector onto the line defined by another vector. — Figure 7.0.27. Orthgonal projection

Source.

Listing 7.0.28. The source for Figure 7.0.27

<diagram dimensions="(300,300)" margins="5">
  <definition>v=(2,1)</definition>
  <definition>b=(2,4)</definition>
  <definition>bhat=dot(v,b)/dot(v,v)* v</definition>
  <definition>bperp=b- bhat</definition>

  <coordinates bbox="[-1,-1,5,5]">
    <grid-axes at="grid-axes"/>

    <line p1="(0,0)" p2="v" infinite="yes" stroke="blue" thickness="2"/>
    <label anchor="(4.4,2.5)">
      <m>L</m>
    </label>

    <label anchor="midpoint(b,bhat)" alignment="ne" clear-background="yes">
      <m>{\mathbf b}^\perp</m>
    </label>
    <vector v="bperp" tail="bhat" stroke="gray"/>

    <vector v="b"/>
    <label anchor="b" alignment="nw">
      <m>{\mathbf b}</m>
    </label>

    <vector v="bhat" stroke="red"/>
    <label anchor="bhat" alignment="se">
      <m>\widehat{\mathbf b}</m>
    </label>

    <vector v="v"/>
    <label anchor="v" alignment="se">
      <m>{\mathbf v}</m>
    </label>

  </coordinates>
</diagram>

A directed network with six nodes and ten edges. — Figure 7.0.29. A simple network

Source.

Listing 7.0.30. The source for Figure 7.0.29

<diagram dimensions="(300,300)" margins="5">
  <definition>graph={1:[3,4,5,5],2:[4,5],3:[4,5],6:[2,4]}</definition>
  <coordinates bbox="(-1,-1,1,1)">
    <network directed="yes" graph="graph" arrows="middle"
             scale="0.8" node-fill="#fcf" seed="1"
             labels="yes" tactile-node-size="40">
    </network>
  </coordinates>
</diagram>

The previous directed network with a loop added and some labels on edges. — Figure 7.0.31. Another network

Source.

Listing 7.0.32. The source for Figure 7.0.31

<diagram dimensions="(300,300)" margins="5">
  <definition>graph={1:[3,4,5,5],2:[4,5],3:[4,5],6:[2,4]}</definition>
  <network directed="yes" graph="graph" scale="0.7"
           node-fill="#fcf" node-stroke="black"
           node-style="circle" node-size="15"
           arrows="end" seed="1" labels="yes"
           tactile-node-size="45">
    <edge vertices="(1,2)">4</edge>
    <edge vertices="(2,2)" label-location="0.3" loop-scale="(0.8,0.8)">
      <m>\alpha</m>
    </edge>
    <node at="2" style="double-circle"/>
  </network>
</diagram>

A Venn diagram showing the difference in two sets. — Figure 7.0.33. A Venn diagram

Source.

Listing 7.0.34. The source for Figure 7.0.33

<diagram dimensions="(300,180)" margins="5">
  <coordinates bbox="(0,0,10,6)">
    <define-shapes>
      <circle at="A" center="(4,3)" radius="2"/>
      <circle at="B" center="(6,3)" radius="2"/>
    </define-shapes>

    <shape shapes="A,B" operation="difference" fill="#fcf" stroke="black"/>
    <shape shape="A" stroke="black"/>
    <shape shape="B" stroke="black"/>
    <rectangle lower-left="(0,0)" dimensions="(10,6)" stroke="black"/>

    <label anchor="(2,3)" alignment="nw"><m>A</m></label>
    <label anchor="(8,3)" alignment="ne"><m>B</m></label>
    <label anchor="(5,0.5)"><m>A\setminus B</m></label>
  </coordinates>
</diagram>

Source.

Listing 7.0.36. The source for Figure 7.0.35

<diagram dimensions="(300,200)" margins="5">
  <coordinates bbox="(0,0,10,9)">
    <path start="(1,1)" stroke="black">
      <lineto point="(1,8)"/>
      <lineto point="(9,8)" decoration="coil; number=5; dimensions=(10,10)"/>
      <lineto point="(9,1)" decoration="capacitor; dimensions=(12,15)"/>
      <lineto point="(1,1)" decoration="zigzag; number=5; dimensions=(10,10)"/>
    </path>

    <point p="(1,4.5)" size="20" fill="white" stroke="black"/>
    <label anchor="(1,4.5)" alignment="nw" offset="(-5,15)" tactile-offset="(-20,20)"><m>+</m></label>
    <label anchor="(1,4.5)" alignment="sw" offset="(-5,-15)" tactile-offset="(-20,-20)"><m>-</m></label>
    <label anchor="(9,4.5)" alignment="west" offset="(-15,0)">
      <m>C</m>
    </label>
    <label anchor="(5,1)" alignment="north" offset="(0,15)">
      <m>R</m>
    </label>
    <label anchor="(5,8)" alignment="south" offset="(0,-15)">
      <m>L</m>
    </label>

  </coordinates>
</diagram>

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