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

2. What is curvature? Refers to how much a geometric object deviates from being “flat” or “straight” The measure of the amount of curving The degree by which a non-linear or surface curves

3. Curvature in Calculus The rate of change of direction of the tangent vector How fast a curve is changing direction Curves that bend to the right, are negative Curves that bend to the left, are positive We will focus on plane curves

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)||

6. Gravity or Curvature? In Euclidian space, gravity in a force that attracts two bodies together In reality, this effect is caused by the curvature of space and time The object is traveling in a strait line Gravity does not redirect the object it redefines what the straightest path is

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

8. How is curvature applied in real life? Applied in mostly in physics and engineering Is used in frictional force Used in calculating space time – orbitals Force = mass x acceleration

9. Example textbook problem Page 876, Section: 12.5, #39 r(t) = 4t i + 3 cos t j + 3 sin t k --- r'(t) = 4 i – 3 sin t j + 3 cos t k T(t) = [r'(t)] / [ II r'(t) II ] = (1/5)[ 4 i – 3 sin t j + 3 cos t k ] T'(t) = (1/5)[ -3 cos t j – 3 sin t k ] K = [ II T'(t) II ] / [ II r'(t) II ] K = (3/5) / 5 K = 3/25

10. Sybase’s Logo

11. The Sybase logo is represented by the Archimedean spiral This is represented by the parametric equations: y = t cos t x = t sin t Insert picture of archimedes.

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:

14. Sources http://www.cs.iastate.edu/~cs577/handouts/curvature.pdf http:// tutorial.math.lamar.edu/classes/calcII/Curvature.aspx http:// www.newworldencyclopedia.org /entry/Curvature http://xahlee.org/SpecialPlaneCurves_dir/ArchimedeanSpiral_dir/archimedeanSpiral.html