Parametric Curve Matching elliptic helix

Aim of task

Student knows how a multidimensional curve is produced from varying a single parameter (Handling mathematical symbols and formalism).
Student can find other parameters of the curve using a 3D visualization (represent mathematical entities, posing and solving mathematical problems, making use of aids and tools ).
Student understands geometrical implications of individual parameters in a multi-dimensional curve (representing mathematical entities).
Student understands that exact geometric features can be approximated numerically (modelling mathematically).


First impression of the question

XML Code
Question description

A 3D curve is plotted. It is a closed curve whose constant parameters are randomly selected upon starting the task. Its equation is

\[t \mapsto \begin{pmatrix} x_0+ a \cdot \cos(t) \cdot \cos(\alpha) - b \cdot \sin(t) \cdot \sin(\alpha) \\ y_0+ a \cdot \cos(t) \cdot \sin(\alpha) + b \cdot \sin(t) \cdot \cos(\alpha) \\ h \cdot t \end{pmatrix}.\]

A second curve given by the same equation is plotted that can be varied using sliders. That way, by matching the two curves, the parameters can be interactively obtained. The task is to find the correct values for $x_0$, $y_0$, $a$, $b$, $\alpha$ and $h$ in real numbers or integer numbers.

Student perspective

The student sees a cartesian coordinate system and a curve plotted in 3D.

It is the task to reconstruct the parameters of the 3D curve. In order to do this they have to find out the radius, amplitude, phase shift and number of oscillations by matching a second curve to the given one using sliders. If they overlap exactly, the parameter values can be read from the sliders and are automatically given as numerical values.


When the student solves the problem

Teacher perspective

The teacher is able to give a list of possible values for parameters. In order to do this, they simply need to modify the entries in the lists specified e.g. change alphar : rand([1/6, 1/4, 1/3, 1/2, 2/3, 3/4, 5/6 ,1]); to alphar : rand([1/6, 1/4, 1/3, 1/2]);.

Another example is the following: change ar: 1+rand(6)/2 to ar: 1+rand(8)/2 in order to select numbers from 1 to 4.5 in steps of 1/2.

For an explanation of the processing of the values read Question variables and Question text.


The above image shows which values the teacher may wish to change

Question code

Question Variables

xr, yr, ar, br and hprogr are randomly selected integer numbers created using rand(), they are divided by 2 and 4 respectively to allow for non-integer steps
alpharis randomly selected from a list of possible values using rand()
The value of alphar is multiplied by pi and saved to alpha as a numerical value. This is done in between numer:true and numer:false.
The value of alphar is multiplied by pi to use it for plotting in Question text.

Question variable code

/*randomize parameters*/

xr:  1+rand(6)/2;
yr:  1+rand(6)/2;
ar: 1+rand(6)/2;
br: 1+rand(6)/2;
hprogr: 1/4+rand(10)/4; 
alphar : rand([1/6, 1/4, 1/3, 1/2, 2/3, 3/4, 5/6 ,1]);


/* prepare numerical values for JSXGraph */
numer:true
alpha: alphar*%pi;
numer:false

/*symbolic data for checking*/
alphar: alphar*%pi

Question Text

Reconstruct the blue 3D curve dependent on a parameter $t$.

In order to do so, you need to find the correct offsets $x_0$ and $y_0$, semi-axis $a$ and $b$, angle of rotation $\alpha$ and amplitude $h$.

The curve is the trace of the function

t \mapsto \begin{pmatrix} 
x_0+ a \cdot \cos(t) \cdot \cos(\alpha) - b \cdot \sin(t) \cdot \sin(\alpha) \\
y_0+ a \cdot \cos(t) \cdot \sin(\alpha) + b \cdot \sin(t) \cdot \cos(\alpha) \\
h \cdot t
\end{pmatrix}.

Use the sliders in the JSXGraph applet or type in your reply right below it.

Task explanation using LaTex
JSXGraph applet using and variables defined in Question variables plotting the 3D curve
[[input:ans1]], [[input:ans2]], [[input:ans3]], [[input:ans4]], [[input:ans5]] and [[input:ans6]] at the end of JSXGraph code to allow input of answers of the student for r, a and phi and n respectively
[[validation:ans1]], [[validation:ans2]] , [[validation:ans3]], [[validation:ans4]], [[validation:ans5]]and [[validation:ans6]] are used for checking of answer

Question text code

<p>Reconstruct the blue 3D curve dependent on a parameter \(t\).</p>
<p>In order to do so, you need to find the correct offsets \(x_0\) and \(y_0\), semi-axis  \(a\) and \(b\), angle of rotation \(\alpha\) and \(z\)-propagation \(h\).</p>
<p> The curve is the trace of the function \[ t \mapsto \begin{pmatrix} 
x_0+ a \cdot \cos(t) \cdot \cos(\alpha) - b \cdot \sin(t) \cdot \sin(\alpha) \\
y_0+ a \cdot \cos(t) \cdot \sin(\alpha) + b \cdot \sin(t) \cdot \cos(\alpha) \\
h \cdot t
\end{pmatrix}.\] </p>
<p>Use the sliders in the JSXGraph applet or type in your reply right below it.</p>

[[jsxgraph width="500px" height="500px" input-ref-ans1='ans1Ref' input-ref-ans2='ans2Ref' input-ref-ans3='ans3Ref' input-ref-ans4='ans4Ref' input-ref-ans5='ans5Ref' input-ref-ans6='ans6Ref']]
var board = JXG.JSXGraph.initBoard(divid,{boundingbox : [-10, 10, 10,-10], axis:false, shownavigation : false});

                              var box = [-5,5];
                             var view = board.create('view3d',
		                            [[-6, -3], [8, 8],
		                            [box, box, box]],
		                            {});

			//curve from STACK values
			var xrad = {#ar#};
			var yrad = {#br#};
			var ang = {#alpha#};
			var xoff = {#xr#};
			var yoff = {#yr#};
                        var hprog = {#hprogr#};
			var c_base = view.create('curve3d', [
				(t) => xoff + xrad* Math.cos(t)* Math.cos(ang) - yrad* Math.sin(t)* Math.sin(ang),
				(t) => yoff + xrad* Math.cos(t)* Math.sin(ang) + yrad* Math.sin(t)* Math.cos(ang),
				(t) => hprog*t,
				[-3* Math.PI,3*Math.PI ]], {strokeWidth: 2, strokeColor: "#1f84bc"});
			
			
				
			//slider-based curve 
			
			var a = board.create('slider', [[-7,-7], [-3,-7], [0,1,10]], {name: "a", snapwidth:0.1, highline: {strokeColor: '#EE442F'}, baseline: {strokeColor: '#EE442F'}});
			var b = board.create('slider', [[-1,-7], [3,-7], [0,3,10]], {name: "b", snapwidth:0.1, highline: {strokeColor: '#EE442F'}, baseline: {strokeColor: '#EE442F'}});
			var y = board.create('slider', [[-1,-6],[3,-6],[-3,2,7]], {name: 'y0', snapwidth:0.1, highline: {strokeColor: '#EE442F'}, baseline: {strokeColor: '#EE442F'}});
			var x = board.create('slider', [[-7,-6],[-3,-6],[-3,2,7]], {name: 'x0', snapwidth:0.1, highline: {strokeColor: '#EE442F'}, baseline: {strokeColor: '#EE442F'}});
			var alpha = board.create('slider', [[-7,-8], [3,-8], [0,1,2* Math.PI]], {name:"alpha", snapwidth:0.05, highline: {strokeColor: '#EE442F'}, baseline: {strokeColor: '#EE442F'}})
			var h = board.create("slider", [[-7,-9], [3,-9], [0,2,5]], {name: "h", snapwidth:0.05, highline: {strokeColor: '#EE442F'}, baseline: {strokeColor: '#EE442F'}})
			
			var c = view.create('curve3d', [
				(t) => x.Value() + a.Value()* Math.cos(t)* Math.cos(alpha.Value()) - b.Value()* Math.sin(t)* Math.sin(alpha.Value()),
				(t) => y.Value() + a.Value()* Math.cos(t)* Math.sin(alpha.Value()) + b.Value()* Math.sin(t)* Math.cos(alpha.Value()),
				(t) => h.Value()*t,
				[-3* Math.PI,3* Math.PI]], {strokeWidth: 2, strokeColor: '#EE442F'}); 

                        stack_jxg.bind_slider(ans1Ref,x);
			stack_jxg.bind_slider(ans2Ref,y);
			stack_jxg.bind_slider(ans3Ref,a);
			stack_jxg.bind_slider(ans4Ref,b);
			stack_jxg.bind_slider(ans5Ref,alpha);
			stack_jxg.bind_slider(ans6Ref,h);

                        /* axis labels*/
                       var xlabel=view.create('point3d',[0.9*box[1],0,(0.6*box[0]+0.4*box[1])], {size:0,name:"x"});
                       var ylabel=view.create('point3d',[0,0.9*box[1],(0.6*box[0]+0.4*box[1])], {size:0,name:"y"});
                       var zlabel=view.create('point3d',[
                           0.7*(0.6*box[0]+0.4*box[1]),
                           0.7*(0.6*box[0]+0.4*box[1]),
                           0.9*box[1]], 
                           {size:0,name:"z"});


[[/ jsxgraph]]
<p>\(x_0=\) [[input:ans1]] [[validation:ans1]]</p>
<p>\(y_0=\) [[input:ans2]] [[validation:ans2]]</p>
<p>\(a=\) [[input:ans3]] [[validation:ans3]]</p>
<p>\(b=\) [[input:ans4]] [[validation:ans4]]</p>
<p>\(\alpha=\) [[input:ans5]] [[validation:ans5]]</p>
<p>\(h=\) [[input:ans6]] [[validation:ans6]]</p>

Answers

Answer ans 1

Answer ans 2

Answer ans 3

Answer ans 4

Answer ans 5

Answer ans 6

Potential response tree

prt1

Feedback variables:

xans: ans1
yans: ans2


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prt2

Feedback variables:

aans: ans3
bans: ans4


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prt3

Feedback variables:

alphaans: ans5
hprogans: ans6


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Aim of task

Question description

Student perspective

Teacher perspective

Question code

Question Variables

Question variable code

Question Text

Question text code

Answers

Answer ans 1

Answer ans 2

Answer ans 3

Answer ans 4

Answer ans 5

Answer ans 6

Potential response tree

prt1

Node 1

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Node 3

prt2

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prt3

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