Projections of the curve onto the coordinate planes
What are the projections of a three-dimensional curve
Sometimes the easiest way to sketch a three-dimensional curve is to sketch its projections on the ???xy???-, ???xz???-, and ???yz???-coordinate planes.
Think about the projections of a curve as the shadows they cast against the coordinate planes.
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You can also think about them as the view of the curve from the coordinate planes. In other words, if you’re standing squarely parallel to the ???xy???-coordinate plane, what you see of the curve is the projection of the curve on the ???xy???-coordinate plane.
Once we have the projections of the curve on each of the coordinate planes, we can use them to draw the three-dimensional graph.
How to find the equation of each projection and then sketch the projections in two dimensions
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Projections of a vector function
Example
Sketch the projections of the curve and use them to sketch the three-dimensional curve.
???r(t)=\langle{t},t^2,t^2+1\rangle???
We’ll convert the vector function to three parametric equations.
???x=t???
???y=t^2???
???z=t^2+1???
To find the projection on the ???xy???-coordinate plane, we need to find an equation in terms of only ???x??? and ???y???, which we’ll do by plugging ???x=t??? into ???y=t^2???.
???y=t^2???
???y=x^2???
We’ll sketch this curve in the ???xy???-coordinate plane.
Think about the projections of a curve as the shadows they cast against the coordinate planes.
To find the projection on the ???xz???-coordinate plane, we need to find an equation in terms of only ???x??? and ???z???, which we’ll do by plugging ???x=t??? into ???z=t^2+1???.
???z=t^2+1???
???z=x^2+1???
We’ll sketch this curve in the ???xz???-coordinate plane.
To find the projection on the ???yz???-coordinate plane, we need to find an equation in terms of only ???y??? and ???z???, which we’ll do by plugging ???y=t^2??? into ???z=t^2+1???.
???z=t^2+1???
???z=y+1???
We’ll sketch this curve in the ???yz???-coordinate plane.
Our final step is to use the projections to sketch the three-dimensional curve. We need a starting point. To find it, we’ll set ???t=0??? in our parametric equations, and get
???x=t???
???x=0???
and
???y=t^2???
???y=0^2???
???y=0???
and
???z=t^2+1???
???z=0^2+1???
???z=1???
Putting these values together, we get the point ???(0,0,1)???. This means that our graph starts at ???(0,0,1)??? and travels upwards in a parabolic shape from the ???xy???- and ???xz???-planar perspective. The three-dimensional graph is
