Vectors in Mathematics anything like vector graphics?

[exdeath]

Right, vector data, meaning it's stored as a series of coordinates for each vertex. The lines between vertices are "vectors". No vector math. And the lines/polygon sides aren't even stored as vectors. They're stored as a beginning and end point. If they were, lines would no longer meet up after reprojecting data.

Maybe I just don't know what vector math is... I remember learning that stuff in algebra and physics and I don't see how any of it relates to vector graphics or geographic vector data.

No. The lines between vertices are line segments known as edges. In 3D graphics the vectors are all between each point and the origin.

 

[exdeath]

You might as well say that raster graphics are like vector math because when you rotate an image trigonometry is used to calculate the new pixels

Yes you could. Each pixel location is a two dimensional vector from some origin.

 

[Throckmorton]

The great thing about Vector graphics is you can scale any sized shape to an unlimited size without losing quality. Vector graphics aren't limited to pixels as far as I can recall. Based on the vocabulary I'm reading, you might not know the difference between vector graphics and raster graphics. As far as I can tell from my Trig class, Vectors aren't limited to any specific number of dimensions either.

What?? I'm saying that based on the logic "trigonometry gets used when vectors are rotated" you could say "trigonometry gets used when rasters are rotated". Doesn't mean rasters have anything to do with vector math.

In other words vector graphics and raster graphics are equally related to vector math.

 

[Throckmorton]

It's amazing that, even in a topic with objective truths, you still try to pull the same FUD you use in P&N. You don't know what you're talking about, so slowly back out of the thread.

Yes, you can describe a curve with a mathematical equation. What does that have to do with vector math? I thought vectors in vector math were all straight, having only a direction and magnitude component...

At the beginning of this thread you said:
I don't know how vector graphics are set up, but based on the name, I always assumed they were based on vector math. Since vector graphics are scalable, they should simply specify direction and relative magnitude of each vector. The relative magnitudes would all then be multiplied by a constant factor for instant scaling.

But vectors in vector graphics and GIS aren't described as a direction and magnitude. Lines are described by a start and end point, curves are stored as equations or the parameters that describe an arc, points by location, polygons by vertices and which side is filled.

 

[Hacp]

Yes, you can describe a curve with a mathematical equation. What does that have to do with vector math? I thought vectors in vector math were all straight, having only a direction and magnitude component...

Hmmmm.. Hmmmmm. Hmmmmm. Hmmmm. No.

 

[Abwx]

The "vectors" in vector graphics aren't the trigonometric operations used to display them in 3D or spin them or whatever display stuff you're talking about. They're the lines and points used to make up the shapes, the defining characteristic of vector graphics, which are unrelated to vector math.

You cant define your line as being simply "two vectors"....
What you call vectors are the extremities/borders/limit
of a mathematical object whose definition is not given
by these "vectors" but by an adequate mathematical
formulae, a first order linear function in the case of your
line or for a flat plane...

If the picture is more complexe, the softare will use
whatever is adequate for the purpose, that is,
polynomials, or whatever transcendental
or complexes functions...

 

[Throckmorton]

You cant define your line as being simply "two vectors"....
What you call vectors are the extremities/borders/limit
of a mathematical object whose definition is not given
by these "vectors" but by an adequate mathematical
formulae, a first order linear function in the case of your
line or for a flat plane...

If the picture is more complexe, the softare will use
whatever is adequate for the purpose, that is,
polynomials, or whatever transcendental
or complexes functions...

Do you mean that in vector graphics a line is always defined by a function?
In GIS a straight line segment is defined only by the vertices. You can reproject or change geographic datums, and a line stays straight because the only thing the software knows is that the line connects the two points.

 

[CycloWizard]

Do you mean that in vector graphics a line is always defined by a function?
In GIS a straight line segment is defined only by the vertices. You can reproject or change geographic datums, and a line stays straight because the only thing the software knows is that the line connects the two points.

Don't you think that GIS software packages probably utilize ellipsoidal coordinate systems to calculate the "straight line?" The two points exist in three-space, so the only "line" between them travels through the earth (with very few exceptions). But I digress - you brought up GIS as a diversion.

 

[exdeath]

Yes, you can describe a curve with a mathematical equation. What does that have to do with vector math? I thought vectors in vector math were all straight, having only a direction and magnitude component...

Ever heard of a vector valued function? f(t) = <x, y, z> ? Vector fields?

f(t) = <cos t, sin t, t> hardly gives you a single straight line with one direction or magnitude.

Finding the direction of your curve at non discrete intervals during interpolation as well as it's surface normals, texture coordinates, etc, all involve vector calculus and linear algebra.

It's ALL vector math.

The simplified "magnitude and direction" definition of a "vector" is 8th grade math.

 

[Throckmorton]

Ever heard of a vector valued function? f(t) = <x, y, z> ? Vector fields?

f(t) = <cos t, sin t, t> hardly gives you a single straight line with one direction or magnitude.

Finding the direction of your curve at non discrete intervals during interpolation as well as it's surface normals, texture coordinates, etc, all involve vector calculus and linear algebra.

It's ALL vector math.

The simplified "magnitude and direction" definition of a "vector" is 8th grade math.

No, I either didn't know about vector valued functions or I forgot.

 

[Fayd]

Ever heard of a vector valued function? f(t) = <x, y, z> ? Vector fields?

f(t) = <cos t, sin t, t> hardly gives you a single straight line with one direction or magnitude.

Finding the direction of your curve at non discrete intervals during interpolation as well as it's surface normals, texture coordinates, etc, all involve vector calculus and linear algebra.

It's ALL vector math.

The simplified "magnitude and direction" definition of a "vector" is 8th grade math.

yeah, once you get to n-dimensional euclidean space, the magnitude and direction definition has completely gone silly. 4th dimensional space might incorporate time..but what does fifth dimensional space incorporate? gravitational effects?

i'm going through my first semester of linear algebra right now. to the op: the vectors you're working with now are nice and logical within 2 and 3 dimensional space. they start getting really abstract to the point that that definition of a vector breaks down.

even so, euclidean space of any dimension is a relatively simple vector space.

no, there was no reason for this post. i included no new information, and have wasted everybody's time.

 

[Bootleg Betty]

yeah, once you get to n-dimensional euclidean space, the magnitude and direction definition has completely gone silly. 4th dimensional space might incorporate time..but what does fifth dimensional space incorporate? gravitational effects?

If you have data about people (for example) with height, weight, sex, income, and how many kids they have. You can then describe each person with five length vector, or as a point in five dimensional space. Euclidean distance between those points can then tell you how "similar" are those people to each other. Some AI algorithms do that.

There are applications in physics too. Imagine a train that can move in one dimension, forward and backwards with a pendulum on top. The easiest way to describe in what state the train is is through coordinate system where one coordinate is position of the train on track, and the other position of the pendulum. The system is then described as a point in two-dimensional space. If you can apply this principle to more complex systems, you can easily get five dimensions or more.

Also there's a way to do you three dimensional math in a way that you transform it to four (or more) dimensional spaces, do you math there in a way that requires less operations and no division, and when you're done, you can switch back to normal space. Aaaaand there's a way to show that vector product in arbitrary space is equivalent to a solution of linear system, but thats black magic.

 

[ncalipari]

I think there are a few errors in this posts:


1) A vector not always represent a point in space, not always we are in a Cartesian coordinate system

2) Not always we have an euclidean space, quite the opposite, we usually use different spaces

3) We never reason by functions, they are too much computationally expensive

4) Linear algebra you study is cool. But is different than the one the computer use. On a CPU you loose associativity, commutativity, denseness, stability, and many other different nice proprieties.

5) Moreover I don't believe in the existence of reals, so if you ask me the situation is even more complex.