Vectors #3: Applications

Today, to finish off the vectors series, I wanted to briefly discuss three applications of vectors to real life problems.

VECTORS IN MOVIES

Vectors can be extremely useful in computer graphics and computer vision, both used when creating computer generated movies. The characters in these movies are modelled as a surface that is made up of connected polygons – usually triangles. The vertices of these polygons are stored into the computers memory in an order such that the vector produced by these vertices is in the direction that points outwards (using the right hand rule, detailed in my previous post).

The outward normal of <i>(A,B,C)</i> is in the opposite direction to <i>(A,C,B)</i> as determined <br>by the right-handed screw rule.

In order to establish the lighting of the scene that is being modelled, the ray tracing algorithm is used.

From the camera’s viewpoint, a ray is traced backwards towards the object, and is reflected off it. If the ray reflects off the facet and intersects the light source, it is shaded in a lighter colour, as in real life it would be lit up by this light source. If not, it is shaded with a darker colour.

The act of tracing the ray backwards is done mathematically using vectors. Suppose the equation of the ray is \[  r = \lambda v  \] and the equation of the plane defined by a facet with vertices a1, a2 and a3 is \[  r = a_1 + \mu _1 (a_2 - a_1) + \mu _2 (a_3 - a_1)  \], to calculate if and where the ray intersects the facet we need to solve these two expressions.

Furthermore, the rotation of three dimensional objects is done using quaternions (discussed in my blog post on Monday). Let’s say we want to rotate a point A with coordinates (a1, a2, a3), through an angle $\beta $, about and axis through the origin, given by a vector b = (b1, b2, b3). First, we construct two quaternions:

\[  q_1 = \cos (\beta /2) + \sin (\beta /2) (b_1 i + b_2 j + b_3 k)  \]

\[  q_2 = \cos (\beta /2) - \sin (\beta /2) (b_1 i + b_2 j + b_3 k) .  \]

Then, using the expression A’ = q1Aq2, we can multiply A by these two quaternions to give the point A’ – the position of A after rotation!

TURBULENCE

Aircraft turbulence is a phenomena that can be explained using Fluid Dynamics, which is a highly mathematical area of science and is of great practical importance. Mathematicians use vector fields to model the speed and direction of a moving fluid through space. A vector field is ‘an assignment of a vector to each point in a subset of space‘. In the case of a velocity field of a moving fluid, a velocity vector is associated to each point in the fluid.

Velocity Field | Source: vis.lbl.gov

Let’s take the example of the velocity of the air. This may vary from point to point and it may also vary with respect to time. This may be shown using the following notation:

u ( x, t )

where u = velocity, x = position in space and t = time.

CLIMATE CHANGE

Vectors can come in useful when studying the movement of ice, which largely concerns the issue of climate change.

Let’s take a piece of ice with an area of 1m2. The net force on the piece of ice arises from the various stresses acting on the ice, such as air stress (due to wind) and water stress (due to currents), as well as the movement of the Earth, which gives rise to the Coriolis force. Combining this gives us:

Force = Acceleration x Mass = Air Stress + Water Stress + Coriolis Force

This can be written vectorially, as only mass is a scalar quantity:

Screen Shot 2016-06-06 at 10.19.42 AM.png

These vectors encode the direction of the movement of the force acting on the piece of ice, as well as the magnitude.

Calculating the values of water stress, air stress and the Coriolis force using individual equations, we can predict where the ice block will move.

Figure 3: The air stress acting on the top of the floe is in the direction of the wind. The water drag is opposite to the direction in which the ice floe is moving. The Coriolis force is at right angles to the direction of movement (shown for the Northern hemisphere). The result is steady motion under a triangle of forces.

In the example above: the air stress acting on the top of the floe (‘a sheet of floating ice’) is in the direction of the wind; the water drag is opposite to the direction in which the ice floe is moving; the Coriolis force is at right angles to the direction of movement (shown for the Northern hemisphere). The result is steady motion under a triangle of forces.

Source: 1 | 2 | 3

Hope you enjoyed this series! M x

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3 comments

  1. I’ve always liked the velocity field as a way of drawing vector-valued functions. It seems to me one of the best ways to link the changes of a system to descriptions, in differential equations, of the system. A well-drawn field almost moves of its own accord.

    Liked by 1 person

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