CURVES Curves in cartesian coordinates Curves in 2D and 3D: explicit, implicit and parametric forms Arc length of a curve Tangent vector of a curve Curves in polar coordinates Conics in polar coordinates Arc length and area under curves in polar coordinates SURFACES Explicit, implicit and parametric forms Area of surfaces CALCULUS III CHAPTER 2: Curves and 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), and reference the dependence of x and y to this variables: Equation form Parametric form Explicit form Implicit form Parametric equations

Curves in 2D Equation form of conic sections Tricks to recognize them Complete the square Why implicit is in general better than explicit?

Parametric form of conic sections (Here t is quoted θ)

Parametrisation of curves

From explicit to parametric form The converse is not always possible: parametric form 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 form:

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 form: 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

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), and reference the dependence of x, y and z to this variables: Equation form Parametric form Explicit form Implicit form 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 form Consider a curve in parametric form

Arc length of a curve in parametric form “Rectification”

Arc length of a curve in equation form 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 form

Curves in polar coordinates CURVES

Polar coordinates (example of curvilinear coordinates) Simple examples  Circle  Straight line

Conics in polar coordinates Relation with implicit form

Arc length of Conics in polar coordinates We use the formula for arc length in parametric form After some calculations

Area of (parametric) Conics in polar coordinates Integration is respect to curvilinear coordinates r, θ (not x,y) Area of r(θ) in polar coordinates

SURFACES

Surfaces - generalities How do we specify a curve? Three main ways Explicit form Implicit form Parametric form

Simple examples of implicit form surfaces

Generalization of conics: quadrics – implicit form

Hipérbola in the plane XZ rotates around Z axis: 1-sheet hyperboloid

London City Hall Catedral de Brasilia Surface of revolution: generated by the rotation of a curve along an axis. 1-sheet hyperboloid

2-sheet hyperboloid

Ellipsoid Deformation of the sphere

Parabola rotate around Z axis Elliptic paraboloid (revolution surface)

Parabola such that its vertes travels another parabola Hyperbolic paraboloid

London City Hall Pringles potato chips are designed using [supercomputing] capabilities to assess their aerodynamic features so that on the manufacturing line they don’t go flying off the line. Dave Turek, vicepresidente del departamento de computación en IBM Hyperbolic paraboloid

Surfaces in parametric form Note that parametrisation of cylinder and sphere coincide with change of variables from cartesian to canonical cylindrical and spherical coordinates respectively

Normal vector to a surface

Area of a surface We can extend the concept of arc length of a curve (a 1D concept) to the area of a surface (2D concept). If the surface is described in explicit form cartesian coordinates

London City Hall Cine: Lost Highway. (Carretera perdida) D. Lynch Moebius Non oriented surfaces Mobius band

London City Hall Non oriented surfaces Klein’s bottle