Parametrized curves - hodographs, supporting surfaces and projections

What curve is represented by the parametric equations $x = t^{2}-t+1$, $y = t^{2}+t+1$?

What curve is represented by the parametric equations $x = t^{2}-2t+3$, $y = t^{2}-2t+1$?

What curve is represented by the parametric equations $x = a\sin^{2}t$, $y = b\cos^{2}t$?

What curve is represented by the parametric equations $x = a\sin^{4}t$, $y = a\cos^{4}t$?

On the curve $x = t^{2}-2t+3$, $y = t^{2}+2t-1$, find the leftmost and the lowest points.

Show that the curve

lies in some plane, and find the equation of this plane.

For what relation among the coefficients $a$, $b$, $c$, $a_{1}$, $b_{1}$, $c_{1}$, $a_{2}$, $b_{2}$, $c_{2}$ does the curve

lie in a plane?

Show that the curve

lies on the surface of a sphere.

Show the same (that the curve lies on the surface of a sphere) for the curve

Convert to parametric form the equation of Viviani's curve, defined by the intersection of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ and the cylinder $x^{2}+y^{2}=ax$, by setting $x=a\sin^{2}t$.

Prove that the projection of the preceding curve onto the plane $xOz$ is a parabola.

Find the projection onto the plane $xOy$ of the curve of intersection of the paraboloid $z=x^{2}+y^{2}$ and the plane $x+y+z=1$.

Find the projection of the curve

onto the plane $xOy$.

Prove that the projection of the helix

onto the plane $yOz$ is a sinusoid.

Prove that, upon translating the origin to the point $O_{1}(0,0,b\beta )$ and rotating the abscissa and ordinate axes about the axis $O_{1}z_{1}$ through an angle $\beta$, the equations of the helix

can be brought to the form

This shows that the helix is capable of sliding along itself.

Construct the hodograph of the vector function $(-\infty < t < +\infty )$:

Prove that the hodograph of the vector function

lies on a sphere.

Prove that the hodograph of the vector function

lies in some plane, and find the equation of this plane.

Prove that the equations

and

are parametrizations of one and the same curve.

Prove that the equations

and

are parametrizations of one and the same curve. How does the point move along this curve as the parameter $t$ increases from $-\infty$ to $+\infty$?

Show that the curve $x = e^{a t} \cos t, y = e^{a t} \sin t, \quad z = e^{a t}$ lies on the cone $z^{2} = x^{2}+y^{2}$.

Show that the curve

$x = t /\left(1+t^{2}+t^{4}\right), \quad y = t^{2} /\left(1+t^{2}+t^{4}\right), \quad z = t^{3} /\left(1+t^{2}+t^{4}\right)$ is a spherical curve.

Under what condition on the matrix $\left(a_{i \jmath }\right)(i, j = 1,2,3)$ is the curve

a planar curve?

Prove that the projection of Viviani's curve

onto the plane of the variables $x$ and $z$ is an arc of a parabola.

Find the projection of the curve $x = e^{t} \sin t, y = e^{t} \cos t, z = t,-\infty < t < +\infty$, onto the plane of the variables $x$ and $y$.

Prove that the projection of the helix

onto the plane of the variables $y$ and $z$ is a sine curve.

Prove that under a translation of the origin to the point $O_{1} = (0 ; 0 ; \beta b)$ and a rotation of the $x$- and $y$-axes about the new $z$-axis through an angle $\beta$, the representation of the helix $x = a \cos t, y = a \sin t$, $z = b t$ can be brought to the form $x_{1} = a \cos t_{1}, y_{1} = a \sin t_{1}, z_{1} = b t_{1}$. This shows that the helix can slide along itself.

Fix two real numbers $a$ and $b, 0 < a < b$. Define a mapping $\mathbf{f} = \left(f_{1}, f_{2}, f_{3}\right)$ of $\mathbb {R}^{2}$ into $\mathbb {R}^{3}$ by

Describe the range $K$ of $\mathbf{f}$. (It is a certain compact subset of $\mathbb {R}^{3}$).