Just wonder where we do this? Worry a little that you are sort of inventing analytic geometry on your own, ad hoc, and the kids would get this later. Or in any case, there are texts that cover this type of transformation in an organized manner, so we could look at these for models/insights. But I admit I don’t remember when this is covered.

]]>You’re doing matrix multiplication rotations through another method.

r00 = Cosine(x) r01 = Sine(x)

r10 = -1 * Sine(x) r11 = Cosine(x)

NewX = x * r00 + y * r10;

NewY = x * r01 + y * r11;

In order to do the first one, you need to subtract 2 from the y coordinate, rotate by 315, flip the y sign,

rotate by 45, and add 2.

For the second one you need to rotate by 247.5 degrees, flip y sign, and rotate by 112.5.

Just working one of them out:

-4, 10

-2 for y:

-4, 8

Sin(315) ~= -.707 or negative (square root of 2 / 2)

Cos(315) ~= .707

-4,8 =

-4*(Sqrt(2)/2) + 8*(Sqrt(2)/2) = 4*(Sqrt(2)/2)

4*(Sqrt(2)/2) + 8*(Sqrt(2)/2) = 12*(Sqrt(2)/2)

Or: 2.828427125,8.485281375

Mirror:

2.828427125,-8.485281375

Or

4*(Sqrt(2)/2), -12*(Sqrt(2)/2)

Sin(45) ~= .707 or again (Sqrt(2)/2)

Cos(45) ~= .707

4*(Sqrt(2)/2) * (Sqrt(2)/2) + -12*(Sqrt(2)/2)* -1*(Sqrt(2)/2)

= 4*(2/4) + (12*(2/4))

= 2 + 6

= 8

4*(Sqrt(2)/2) * (Sqrt(2)/2) + -12*(Sqrt(2)/2)* (Sqrt(2)/2)

= 4*(2/4) + (-12*(2/4))

= 2 – 6

= -4

8,-4

+2

= 8,-2

So this one was nice because you’re multiplying the square root of 2 with itself and some other things.

————-

The rotation for the second one is a great deal messier. the angle is 90+22.5 = 112.5

Cos(247.5) = -.382683432

Cos(112.5) = -.382683432

Sin(247.5) = -.923879533

Sin(112.5) = .923879533

not doing the math for this one.

This by contrast is apparently the square root of 0.146446609 and the square root of 0.853553391 and they aren’t canceling out.

]]>Dear Jim:

Thank you for your comment.

Also:

To solve equation x+5=3,

one had to introduce new numbers: negative ones.

To solve equation x*7=6,

one had to introduce new numbers: fractions, or rational numbers.

To solve equation x^2=2/3,

one had to introduce new numbers: irrational ones.

Skip real transcendental numbers, like e or pi.

To solve equation x^2 +1 = 0,

one had to introduce new numbers: complex ones.

However, to solve equation x^2 – i = 0,

one _did_not_have_ to introduce new numbers like “hypercomplex”:

x=(1+i)/sqrt(2) does the job.

Any polynomial has a root within the field of complex numbers

(Fundamental Theorem of Algebra).

Shout it from the rooftops !

With invariable respect of educationrealist,

your F.r.

Heavens. This is where the fact that I’m not a mathematician comes into play. But thanks, I’ll read up.

]]>Also in physics the role that orientation plays in the basic structure of the universe is a fascinating topic.

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