Question

Q. What are the projections of helix (acost, asint, t) in all three coordinate planes xy-plane, yz-plane, xz-plane.

Answer 1

Answer on Question #70299 – Math – Analytic Geometry

Question

What are projections of helix (acost, asint, t) in all three coordinate planes xy-plane, yz-plane, xz-plane?

Solution

1) projection of helix (acost, asint, t) in xy-plane is (acost, asint). The curve is $x = \text{acost}$, $y = \text{asint}$. If we take $x^2 + y^2$, then we get $x^2 + y^2 = (\text{acost})^2 + (\text{asint})^2 = a^2((\text{cost})^2 + (\text{sint})^2) = a^2$. This is a circle of radius $a$: $x^2 + y^2 = a^2$.

2) projection of helix (acost, asint, t) in xz-plane is (acost, t). The curve is $x = \text{acost}$, $z = t$. So, $x = \text{acosz}$. This is a cosine function: $x = \text{acosz}$.

3) projection of helix (acost, asint, t) in yz-plane is (asint, t). The curve is $y = \text{asint}$, $z = t$. So, $y = \text{asinz}$. This is a sine function: $y = \text{asinz}$.

**Answer**: $x^2 + y^2 = a^2$, $x = \text{acosz}$, $y = \text{asinz}$.

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