# vector to scalar mirror

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Hi.

I have a rather strange wondering.

If you have a vector V= (a,b,c,d) of four scalars, and a scalar "twin" of V which is S = a<<24 + b<<16 + c<<8 + d (the abcd become bytes of 32bit unsigned integer), then, what scalar operation s(x) on S needs to be done so that resulting scalar R would be "twin" of vector (x*a,x*b,x*c,x*d).

Like:

V= (a,b,c,d) => S= a<<24 + b<<16 + c<<8 + d;

s(x,S)=R <=> R= (a*x)<<24 + (b*x)<<16 + (c*x)<<8 + d*x;

thus

s(x,S)=(a*x)<<24 + (b*x)<<16 + (c*x)<<8 + d*x; for a scalar S=a<<24 + b<<16 + c<<8 + d derived from twin vector V (a,b,c,d)

last question, is following sentence true?

for a vector V is always one scalar S and for a scalar S is allways one vector V? (I think yes)

final exact formulation:

"V transformed by matrix (2^24 , 2^16 , 2^8 , 1) results in scalar h

and

V*x transformed by matrix (2^24 , 2^16 , 2^8 , 1) result in scalar o(h) , what is definition of o function, if the function exists?"

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Assuming I understood the intentions, I don't believe there is such a thing. Putting my understanding of your question in simplistic terms without the fancy nomenclature, I think what you are talking about is the classic compression of a set of values into a packed form. Ignoring floats and working on let's say a 32 bit color with 8 bit components, your first example would be:

uint8_t r=255, g=255, b=255, a=255; // RGBA = Solid White.
uint32_t packed = r<<24 + g<<16 + b<<8 + a; // "Packed" value.

So now adding two packed values is one of the goals. There is no fundamental problem with taking 0xF1F1F1FF and adding 0x01010100 together, everything will work out correctly and end up as 0xF2F2F2FF properly. Unfortunately the big problem is that if you tried adding 0x0f0f0f00 the result would be completely random. (Well very specific but very random for your desired results.)

So, while it is provably possible to come up with the proper item to add to build the proper result, it will cost as much, if not more, time to figure it out. Given the fact that each "byte" is supposed to work in a 0-255 range the only byte which will act normally is the 'a' byte in the given example. It will wrap properly and everything is great, unfortunately the b byte will catch the result of the a byte wrap and all that means only a works correctly. In order to prevent a from screwing up b, you have to break things back into the bytes to figure out if they will wrap. As such, there "is" an answer, but unlikely to be much of a benefit for most purposes.

NOTE: thinking about this, I can see some ideas to compress things for network compression, but otherwise not sure what purpose you are looking for.

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Watch out with the precedence of << (it is lower than +).

The answer (assuming you want each value to wrap around) is

v' = ( ((((a>>24) * x ) & 255) << 24) + ((((b>>16) * x ) & 255) << 16) + ((((c>>8) * x ) & 255) << 8) + ((d * x) & 255) )

which isn't very helpful (replace & 255 with mod 256 is you want to use the mathematical term).

the other question you ask is if the operation is one to one (i.e. bijective, therefore invertible), the answer is no unless 256 and (x mod 256) are coprime, e.g. (128 * 2) mod 256 = (64 * 4) mod 256 = 0 mod 256

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yes, forgot to mention the vectors scalars exist on (0-255) interval body, and body modulates (254+3=1).

You may say it would be not effective to use scalar representant to operate on, but in case of high expense of memory access, that would not be the case. But it seems impossible since modulation leftover would travel onto more prior  byte-memeber, if scalar operation would apply. Thus, we can conclude this impossible.

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