Math 252 Lecture 08: Wednesday, January 20th 1999.

We started off [Transparancy 1] by considering the velocity vector when the magnitude of the position vector stays constant. It was shown that the velocity vector is perpendicular to the position vector under these circumstances.

We then looked in some detail at circles in space [Transparancy 2], coming up with a general formula for an arbitrarily oriented circle in 3-space.

An example was worked out, in detail, of how to find the vector equation of a circle cutting the axes at x=1, y=1, and z=1. [Transparancies 3 to 5.] Use was made of one of the results worked out in Homework Assignment #1, which enabled us to find the centre of the circle and its corresponding radius.

Next, we considered the vector equation of an arbitrarily oriented helix in 3-space [Transparancy 6]. The previous example, of a circle cutting the axes at x=1, y=1, z=1, was extended into a helix of the same radius oriented along the same axis. The equation was plotted in Maple, and here's what we obtained:

> x:=1/3+sqrt(3)/3*cos(t)+1/3*sin(t)+t/10/Pi;

[Maple Math]

> y:=1/3-sqrt(3)/3+1/3*sin(t)+t/10/Pi;

[Maple Math]

> z:=1/3-2/3*sin(t)+t/10/Pi;

[Maple Math]

> with(plots):

> spacecurve([x,y,z],t=0..12*Pi,axes=NORMAL,scaling=CONSTRAINED,numpoints=1000,orientation=[-25,70]);

[Maple Plot]

This graph agrees with what we expected.

By interpreting the paramater "t" as time, one can consider the velocity vector of a spacecurve [Transparancy 7]. This gives us an expression for the Unit Tangent Vector to the spacecurve.

Finally, by approximating a spacecurve by a large number of short segments, and taking the limit of an infinite number of infinitesimally small segments, one finds an integral for the arc length along the spacecurve [Transparancies 8 and 9]. The space curve can, in principle, be re-paramaterized in terms of arc length rather than "t", but this is usually too complicated to do.

Here are the scanned-in transparancies, in full-colour JPEG format:

(You will find two versions -- a "screen" version, which is 400 pixels wide and a corresponding number of pixels long, at a resolution of 100 by 100 dots per inch, and a "print" version, which is generally between 8 and 8.5 inches wide and up to 11 inches long, also at a resolution of 100 by 100 dots per inch. The screen versions generally have a file size of between 60 and 75 K, which downloads reltively quickly. The print versions generally have a file size of between 170 and 225 K, which takes a bit longer to download.)

SFU / Math & Stats / ~hebron / math252 / lec_notes / lec08/index.html
Revised January 1999 by John Hebron.