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→Challenging Problems
==Challenging Problems==
==Challenging Problems==
Not only lines can be described with equations like <math>x=x(t), y=y(t), z=z(t) \ </math>. When the coordinates are not linear functions of <math>t \ </math>, these equations describe a curve. These kinds of equations are called parametric, because instead of describing shapes with only the coordinates <math>x,y,z \ </math>, they use a '''parameter,''' in this case, <math>t \ </math>. We'll discuss this more in later lectures.
'''1.''' Describe the curve <math>x=\cos(t), y=\sin(t), z=t \ </math>. What shape does this curve have?
'''2.''' Curves are frequently encountered when two surfaces intersect. Do the surfaces <math>z=x^2-y^2 \ </math> and <math>x^2 + y^2 - z^2 = 1 \ </math> intersect? Can you give a parametric description of the curve of intersection??