Tag: Geometry

Geospatial Slope Estimation

Geospatial Slope Estimation

Geospatial Slope Estimation

08/23/20

“How does geospatial software estimate the slope of maps?”

Slope analysis is extremely useful for numerous geospatial applications ranging from housing placement to wildfire risk estimation. However, how exactly is the sloe calculated? Well, what geospatial software usually does is take each data point of height and compare it to its eight neighbors to estimate the height difference. This is how Geospatial Slope Estimation is performed.

Image credit http://wiki.gis.com/

Hypersphere

Hypersphere

Hypersphere

11/12/16

“What does a four-dimensional sphere look like?”

 

Let’s think about something. A circle can be classified as an object in 2-dimensional space whose boundary is composed of all of the points equidistant from a center point, where the radius is composed of two cartesian coordinates ( r=sqrt(x^2+y^2)) and the area is proportional to the square of the radius (A=pi*r^2). A sphere in turn is the 3-dimensional version of this, where all of the points equidistant from a center makes up the object, with the radius being composed three cartesian coordinates and the volume is proportional to the cube of the cube of the radius (V=4/3 * pi * r^3 ), and if were to take a two-dimensional cross-section, we would obtain a circle. .But what if we were to take this concept into higher dimensions? Well, let’s explore the concept. A four-dimensional sphere (termed a hypersphere by mathematicians) would have to be described with four cartesian coordinates, and the “hyper-volume” would be proportional to the fourth power of the radius (V= ½ * pi^2 * r^4),and if were to take a three dimensional cross section of a hypersphere, we would find a regular three-dimensional  sphere. 

Surface area

Surface area

Surface area

10/30/16

“How can we quantize the exterior surfaces of geometric objects?”

 

It is safe to say that almost every human individual can easily relate to three-dimensional objects. However, mathematicians do not feel suffice with just simple qualitative descriptions of space, but rather they seek to go deeper, into the quantitative realm. And as such, they will take processes and patterns that we see everyday and systematize them in a rigorous manner. One of the properties that mathematicians will analyze include the surface area of a three dimensional objects. To put it in simple terms, the surface area of an object is the outer layer that envelops it (think of something like skin on a human). Measuring the surface area has many practical applications in the natural sciences. For example, in physics we can use the surface area of a Gaussian surface to measure the electric field due in a certain area with Gauss law, and in biology by using the surface area to volume ratio of a cell membrane to quantize how rapidly a substance will spread from the interior the the outer coating. All in all, surface area is a fascinating concept with numerous applications to the real world.

Polyhedra

Polyhedra

Polyhedra

09/28/16

“How do we classify geometric objects with flat faces?”

 

Think of an object in three dimensions. Any object. You can probably come up with a wide array of diverse and seemingly unending chaotic shapes. But in geometry, we need to organize everything into patterns with special properties. So since we have already started with three dimensional objects, let’s go classify the simplest possible three-dimensional objects. But what exactly are they? Well, let’s think about it. We know that two dimensional objects are simply flat faces. So what if we took many of those flat faces and strung them together in an organized manner? This is the primary principle behind polyhedrons. Polyhedrons are three dimensional objects made completely out of two dimensional faces, with no opening. Did you know that every day objects such as cubes and pyramids are polyhedra?

Vertices

Vertices

Vertices

09/27/16

“How can we classify the point where two lines meet?”

 

When working with Geometry, We will have to deal with alot of different and divergent phenomena. In order to organize all of this for practical use, we will have to give such special phenomena special terminology. So let’s do an example with the intersection of two lines.. Geometers (Mathematicians who research geometry) have decided to name this intersection a vertex. Since all vertices involve the intersection of two lines, vertices always include an angle.
But let’s not limit ourselves to just simple lines, let’s look at an interesting application of vertexes to more complex three dimensional structures. When two vertices meet in a polyhedron, a mathematical object called an edge is formed. But what is even more fascinating is that if you take all of the faces (the 2-dimensional shapes that make up the outer surface) of a polyhedron, and add the difference of the Vertices and angles, the answer will always equal two! We can represent this symbolically as F+E-V = 2