Curvature refers to how much a geometric object deviates from being flat or straight. It is a measure of the amount of curving or bending of a curve or surface. In calculus, curvature is defined as the rate of change of the direction of the tangent vector to a curve as it moves along the curve. Curvature plays an important role in physics and engineering, where it is used to describe concepts like gravitational acceleration and frictional forces.

1. Curvature Jasmine He Victoria de Metz Rashi Ojha

4. Curvature equation K-Curvature S-Arc Length -Angle measured counterclockwise T-

5. Proof R(θ )=rcosθ I + rsinθ j R(s)=rcos(s/r) I + rsin (s/r) j R’(s)=-sin(s/r) I + cos (s/r) J T(s) = r’(s)/||r’(s)||=-sin(s/r) i +cos(s/r) J K= ||T’(s)||

7. Relationship between Acceleration, Speed, and Curvature a(t) = d2s T + K (ds/dt)^2 N dt2 K- Curvature ds/dt- Speed

10. Sybase’s Logo

12. Curvature of the Sybase Logo Archimedes’ Spiral Parametric Equation: x = t cos t y = t sin t K = ( Iy''I )/ [1 + (y')²]³∕² y' = [( t cos t) + sin t] / [ cos t – t sin t] y'' = (d/dt [ (sin t + t cos t) / ( cos t – t sin t)]) · (1 / dx/dt ) y'' = [ (cos t + cos t – t sin t)(cos t – t sin t) – ( sin t + t cos t)(-sin t – sin t – t cos t) ] / (cos t – t sin t)³ y'' = [ ( 2 cos t – t sin t)(cos t – t sin t) + ( sin t + t cos t)(2 sin t + t cos t)] / (cos t – t sin t)³ y'' = [ (2 cos²t – 2t sin t cos t – t sin t cos t + t² sin²t) + (2 sin²t + 2t sin t cos t + t sin t cos t + t² cos²t) ] / (cos t – t sin t)³ y'' = (2 + t²) / ( cos t – t sin t)³

13. Curvature of the Sybase Logo cont. Finding the curvature: K = I y'' I / [ 1 + (y')²]³∕² K = I (2 + t²) / ( cos t – t sin t)³ I / ([ 1 + ([( t cos t) + sin t] / [ cos t – t sin t])²]³∕² K = (2 + t²) / [ 1 + (t cos t + sin t)²]³∕² Graph the curvature: