Kenneth Kuttler

Chapter 15

Motion On A Space Curve

15.1 Space CurvesA fly buzzing around the room, a person riding a roller coaster, and a satellite orbiting theearth all have something in common. They are moving over some sort of curve in threedimensions.

Denote by R(t) the position vector of the point on the curve which occurs at time t.Assume that R′,R′′ exist and are continuous. Thus R′ = v, the velocity and R′′ = a isdefined as the acceleration.

R(t)

Lemma 15.1.1 Define T (t) ≡ R′ (t)/∣∣R′ (t)∣∣. Then |T (t)| = 1 and if T ′ (t) ̸= 0, then

there exists a unit vector N (t) perpendicular to T (t) and a scalar valued function κ (t),with T ′ (t) = κ (t) |v|N (t).

Proof: It follows from the definition that |T | = 1. Therefore, T ·T = 1 and so, upondifferentiating both sides,

T ′ ·T +T ·T ′ = 2T ′ ·T = 0.

Therefore, T ′ is perpendicular to T . Let N (t)∣∣T ′∣∣≡ T ′. Note that if

∣∣T ′∣∣= 0, you couldlet N (t) be any unit vector. Then letting κ (t) be defined such that

∣∣T ′∣∣ ≡ κ (t) |v (t)|, itfollows

T ′ (t) =∣∣T ′ (t)∣∣N (t) = κ (t) |v (t)|N (t) . ■

Definition 15.1.2 The vector T (t) is called the unit tangent vector and the vector N (t) iscalled the principal normal. The function κ (t) in the above lemma is called the curvature.The radius of curvature is defined as ρ = 1/κ . The plane determined by the two vectors T

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Chapter 15Motion On A Space Curve15.1 Space CurvesA fly buzzing around the room, a person riding a roller coaster, and a satellite orbiting theearth all have something in common. They are moving over some sort of curve in threedimensions.Denote by R(t) the position vector of the point on the curve which occurs at time f.Assume that R’,R” exist and are continuous. Thus R’ = v, the velocity and R” = a isdefined as the acceleration.R(t)xLemma 15.1.1 Define T(t) = R'(t)/|R'(t)|. Then |T (t)| = 1 and if T'(t) #0, thenthere exists a unit vector N (t) perpendicular to T (t) and a scalar valued function « (t),with T’ (t) = K(t)|v| N (2).Proof: It follows from the definition that |T'| = 1. Therefore, T-T = 1 and so, upondifferentiating both sides,T’-T+T-T' =2T'-T =0.Therefore, T” is perpendicular to T’. Let N (r) |T"| = 7". Note that if |7”| =0, you couldlet IN (t) be any unit vector. Then letting «(t) be defined such that |T"| = K(f) |v(s)], itfollowsT(t) =|P()|NOQ=KOloOINO.MDefinition 15.1.2 The vector T (t) is called the unit tangent vector and the vector N (t) iscalled the principal normal. The function k(t) in the above lemma is called the curvature.The radius of curvature is defined as p = 1/«. The plane determined by the two vectors T275