CURVES Curves in cartesian coordinates Curves in 2D in addition to 3D: explicit, implicit a

CURVES Curves in cartesian coordinates Curves in 2D in addition to 3D: explicit, implicit a

David, Paul, News Director has reference to this Academic Journal, PHwiki organized this Journal CURVES Curves in cartesian coordinates Curves in 2D in addition to 3D: explicit, implicit in addition to parametric as long as ms Arc length of a curve Tangent vector of a curve Curves in polar coordinates Conics in polar coordinates Arc length in addition to area under curves in polar coordinates SURFACES Explicit, implicit in addition to parametric as long as ms Area of surfaces CALCULUS III CHAPTER 2: Curves in addition to surfaces Curves in cartesian coordinates CURVES Curves in 2D Curves in 2D are geometric shapes that can be mathematically characterized in several ways. In cartesian coordinates (x,y), curves are described by: or The third way is to parametrise the curve using an additional variable (the parameter, usually t), in addition to reference the dependence of x in addition to y to this variables: Equation as long as m Parametric as long as m Explicit as long as m Implicit as long as m Parametric equations

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Curves in 2D Equation as long as m of conic sections Tricks to recognize them Complete the square Why implicit is in general better than explicit Parametric as long as m of conic sections (Here t is quoted ) Parametrisation of curves

Parametrisation of curves From explicit to parametric as long as m The converse is not always possible: parametric as long as m is more general. Simple examples Examples of parametric curves in 2D Cycloid: Describes the trajectory of a circles point, when the circle rotates along a straight line. Parametric as long as m: What if the rotating point does not belong to the circle Examples of parametric curves in 2D

Trochoid: The rotating point does not belong to the circle. Train wheel Parametric as long as m: circles radius, distance of the point to the center Examples of parametric curves in 2D Epicicloids: The circle rotates along another circle. Examples of parametric curves in 2D Epitrocoids: Same as epicicloids, but the point from which the trajectory is calculated does not belong to the circle. Examples of parametric curves in 2D

Wankel engine: epicicloid Mazda RX8 Hipocicloids: Examples of parametric curves in 2D Catenary: is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends. The curve has a U-like shape, superficially similar in appearance to a parabola (though mathematically quite different) Galileo thought that it was a parabola. Huygens (1650) proved when he was 17 that it wasnt a parabola, although he couldnt find the correct equation. Bernoulli found it in 1691using physical considerations. Examples of parametric curves in 2D

Examples of parametric curves in 2D Lissajous figures (trajectory that develops from coupled oscillators) The trajectory is closed if k1/k2 is rational Curves in 3D Curves in 3D are geometric shapes that can be mathematically characterized in several ways. In cartesian coordinates (x,y,z), curves are described by: or The third way is to parametrise the curve using an additional variable (the parameter, usually t), in addition to reference the dependence of x, y in addition to z to this variables: Equation as long as m Parametric as long as m Explicit as long as m Implicit as long as m z=f(x,y) V(x,y,z)=0

Helix DNA chains are coupled helix Examples of parametric curves in 3D Eudoxus Hypopedes: intersection of a sphere with a cone whose axis is tangent to the sphere Application: tennis Examples of parametric curves in 3D Arc length of a curve Rectification Length L

Arc length of a curve in parametric as long as m Consider a curve in parametric as long as m Arc length of a curve in parametric as long as m Rectification Arc length of a curve in equation as long as m Curves in 2D y=f(x) Curves in 3D z=f(x,y) Pythagoras

Parametrisation of a curve y=f(x) using its arc of length Tangent vector of a curve Consider a curve in parametric as long as m Curves in polar coordinates CURVES

Polar coordinates (example of curvilinear coordinates) Simple examples Circle Straight line Conics in polar coordinates Relation with implicit as long as m Arc length of Conics in polar coordinates We use the as long as mula as long as arc length in parametric as long as m After some calculations

London City Hall Cine: Lost Highway. (Carretera perdida) D. Lynch Moebius Non oriented surfaces Mobius b in addition to London City Hall Non oriented surfaces Kleins bottle

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