the family of tangent lines to a spacecurve
We use the twisted cubic as an example. At each point along the curve (parameter t ), we can write the vector-valued function representing the tangent line, which itself has an independent parameter (say s ). By plotting the function of the two parameters LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYlLUYjNictSSNtaUdGJDYmUSJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUSJzRidGNEY3RjpGN0ZBRjdGQUY3RkE= , locating where we are along the curve, and where we are along its tangent line there, we can see a representative selection of the tangent lines along the curve. If we choose a single point along the curve we can display the curve and tangent line there together.
For this example, the x -axis is the tangent line at the origin LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjBGJ0YyRjxGMkY8 (cyan) and the straight lines in the surface grid at the end of this worksheet are the other tangent lines.
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
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyNYKiUhR0YoRihbZ2whIyUhISEiJCIkLUklbXJvd0c2JEYmL0krbW9kdWxlbmFtZUdGKEYkNiUtSSNtaUdGMDYjUSJ0RigtSSNtb0dGMDYjUScmcGx1cztGKC1GNTYjUSJzRigtRi82JS1GLzYnLUkjbW5HRjA2I1EiMkYoLUY5NiNRMSZJbnZpc2libGVUaW1lcztGKEY8RkdGNEY4LUklbXN1cEdGMDYkRjRGQy1GLzYlLUYvNictRkQ2I1EiM0YoRkdGPEZHRkpGOC1GSzYkRjRGUTcjLSZJJ1ZlY3RvckdGJTYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1b3V5XFdCIVwkKm8k
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyNYKiUhR0YoRihbZ2whIyUhISEiJCIkLUkjbWlHNiRGJi9JK21vZHVsZW5hbWVHRihGJDYjUSJzRigtSSNtbkdGMDYjUSIwRihGNTcjLSZJJ1ZlY3RvckdGJTYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCI1X0t5XFdCIVwkKm8k
Here is the display of the curve and its tangent line at the origin. By using the same interval (in this case LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEoJiM5MTY7dEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjFGJ0YyRjxGMkY8 ) about the t value for the point of tangency for the tan line interval we get the point of tangency centered in the plot, namely: 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 First we do a point not at the origin LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjFGJ0YyRjxGMkY8 :
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For zooming purposes we use the tangent line at the origin.
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Dragging the slider to the left zooms in on the tangent line at the origin, close to which the parameter s along the tangent line corresponds closer and closer to the parameter t on the curve.
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SSJURzYi #
# The following code is autogenerated by the Explore command. Do not edit.
#
Explore:-Runtime:-Update( plots:-display(plots:-spacecurve(R(t),t = VectorCalculus:-`-`(T) .. T,color = red,thickness = 2),plots:-spacecurve(VectorCalculus:-`+`(R(0),VectorCalculus:-`*`(s,eval(VectorCalculus:-diff(R(x),x),x = 0))),s = VectorCalculus:-`-`(T) .. T,color = blue,thickness = 2),scaling = constrained), ':-Plot0', [':-T'], ':-sliders'=["SliderT"], ':-labels'=["LabelT"],':-isplot'=true, ':-handlers'=[T=handlertab[T]]);
#
# End of autogenerated code.
#