Math 252 Lecture 09: Friday, January 22nd 1999.

We started looking at acceleration and curvature along a spacecurve (section 2.3 of the textbook). Acceleration has two components [Transparancy 1], tangential and centripetal. The latter is defined in terms of a radius of curvature. The curvature constant "k" is defined as the magnitude of the derivative of the unit tangent vector with respect to arc length. The meaning of "k" is explored geometrically [Transparancy 2], and it is found to be the inverse of the radius of curvature.

The principal normal vector (N), is defined [Transparancy 3] as a unit vector perpendicular to the tangent vector, and pointing along the radius of curvature towards the centre of the osculating circle. (This is the direction of the derivative of the tangent vector with respect to arc length.)

The acceleration vector is decomposed in terms of the unit tangent vector (T) and the unit normal vector (N), and an expression for the tangential and centripetal acceleration is found [Transparancy 4].

Frenet Formulas are introduced [Transparancy 5] and the binormal vector is defined.

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 / lec09/index.html
Revised January 1999 by John Hebron.