10/8/2002, CURVES Math 21a, O. KnillGRAPHS. If f(x) is a function of one variable,then {(x, f(x))} is a graph. Graphs are exam-ples of curves in the plane but form a rathernarrow class of curves.EXAMPLE 1. Let f(x) = cos(x) + x sin(x)defined on [−π, π]. The graph of f is shownin the picture to the right.-3 3xf(x)PARAMETRIC CURVES. If f(t), g(t) are functions in one variable, defined on some parameter intervalI = [a, b], then r(t) = (f(t), g(t)) is a parametric curve in the plane. The functions f(t), g(t) are calledcoordinate functions. Often, especially in physical context, one write x(t) = f(t) and y(t) = g(t).EXAMPLE 2. If x(t) = t, y(t) = t2+ 1, we can write y(x) = x2+ 1 and the curve is a graph.EXAMPLE 3. If x(t) = cos(t), y(t) = sin(t), then r(t) is a circle.If x(t), y(t), z(t) are functions, then r(t) = (x(t), y(t), z(t)) describes a curve in space.EXAMPLE 4. If x(t) = cos(t), y(t) = sin(t), z(t) = t, then r(t) describes a spiral.IDEA: Think of the parameter t as time. For every fixed t, we have a point (x(t), y(t), z(t)) in space. As tvaries, we move along the curve.EXAMPLE 5. If x(t) = cos(2t), y(t) = sin(2t), z(t) = 2t, then we have the same curve as in EXAMPLE 4 butwe traverse it faster. The parametrisation changed.EXAMPLE 6. If x(t) = cos(−t), y(t) = sin(−t), z(t) = −t, then we have the same curve as in EXAMPLE 4but we traverse it in the opposite direction.EXAMPLE 7. If P = (a, b, c) and Q = (u, v, w) are points in space, then r(t) = (a+t(u−a), b+t(v−b), c+t(w−c))defined on t ∈ [0, 1] is a line segment connecting P with Q.ELIMINATION: Sometimes, it is possible to eliminate the variable t and write the curve using equations (oneequation in the plane or two equations in space).EXAMPLE: (circle) If x(t) = cos(t), y(t) = sin(t), then x(t)2+ y(t)2= 1.EXAMPLE: (spiral) If x(t) = cos(t), y(t) = sin(t), z(t) = t, then x = cos(z), y = sin(z). The spiral is theintersection of two graphs x = cos(z) and y = sin(z).CIRCLE HEART LISSAJOUS SPIRAL(cos(t), 3 sin(t)) (1 + cos(t))(cos(t), sin(t)) (cos(3t), sin(5t))et/10(cos(t), sin(t))TRIFOLIUM EPICYCLESPRING (3D) TORAL KNOT (3D)− cos(3t)(cos(t), sin(t))(cos(t) + cos(3t)/2, sin(t) +sin(3t)/2)(cos(t), sin(t), t)(cos(t) + cos(9t)/2, sin(t) +cos(9t)/2, sin(9t)/2)WHY DO WE LOOK AT CURVES?Particles, bodies, quantities changing in time. Examples: motion of a starin a galaxy. Data changing in time like (DJIA(t),NASDAQ(t),SP500(t))Strings or knots are curves which are important in theoretical physics. Knots are closed curves in space.Complicated molecules like RNA or proteines can be modeled as curves.Computergraphics: surfaces are represented by mesh of curves. Representing smooth curves efficiently isimportant for fast rendering of scenes. The progress in this field is not only due to better computers but alsodue to mathematics.Space time A curve in space-time describes the motion of particles. In general relativity gravity is describedthrough curves in a curved space time.Curves are also interesting in topology (i.e. peano curves, boundaries ofsurfaces, knots).POLAR COORDINATES. A point (x, y) inthe plane has the polar coordinates r =px2+ y2, θ = arctg(y/x). We have x =r cos(θ), y = r sin(θ).xyP=(x,y)=(r cos(t),r sin(t))O=(0,0)r=d(P,O)tPOLAR CURVES. A general polar curve is written as (r(t), θ(t)). Itcan be translated into x, y coordinates: x(t) = r(t) cos(θ(t), y(t) =r(t) sin(θ(t)).POLAR GRAPHS. Curves which are graphs when written in polar co-ordinates are called polar graphs.EXAMPLE 8. r(θ) = cos(3θ) is the trifoil which belongs to the classof roses r(t) = cos(nt).EXAMPLE 9. If y = 2x + 3 is a line, then the equation gives r sin(θ) =2r cos(θ) + 3. Solving for r(t) gives r(θ) = 3/(sin(θ − 2 cos(θ)). The lineis also a polar
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