Activity 3.3.2
In this activity, we will explore the difference between these two coordinate systems.

First, we will explore the \(RGB\) color model.
COLORRGB
What happens when \(G=0\text{,}\) \(B=0\) (pushed all the way to the left), and \(R\) is allowed to vary?
What happens when \(R=0\text{,}\) \(G=0\text{,}\) and \(B\) is allowed to vary?
How can you create black in this color model?
How can you create white?

Next, we will explore the \(YC_bC_r\) color model.
COLORYCbCr
What happens when \(C_b=0\) and \(C_r=0\) (kept in the center) and \(Y\) is allowed to vary?
What happens when \(Y=0\) (pushed to the left), \(C_r=0\) (kept in the center), and \(C_b\) is allowed to increase between 0 and 127.5?
What happens when \(Y=0\text{,}\) \(C_b=0\text{,}\) and \(C_r\) is allowed to increase between 0 and 127.5?
How can you create black in this color model?
How can you create white?
Verify that \(\bcal\) is a basis for \(\real^3\text{.}\)
Find the matrix \(C_{\bcal}\) that converts from \(\threevec{Y}{C_b}{C_r}\) coordinates into \(\threevec{R}{G}{B}\) coordinates. Then find the matrix \(C_{\bcal}^{1}\) that converts from \(\threevec{R}{G}{B}\) coordinates back into \(\threevec{Y}{C_b}{C_r}\) coordinates.

Find the \(\threevec{Y}{C_b}{C_r}\) coordinates for the following colors and check, using the diagrams above, that the two representations agree.
Pure red is \(\threevec{R}{G}{B}= \threevec{255}{0}{0}\text{.}\)
Pure green is \(\threevec{R}{G}{B}= \threevec{0}{255}{0}\text{.}\)
Pure blue is \(\threevec{R}{G}{B}= \threevec{0}{0}{255}\text{.}\)
Pure white is \(\threevec{R}{G}{B}= \threevec{255}{255}{255}\text{.}\)
Pure black is \(\threevec{R}{G}{B}= \threevec{0}{0}{0}\text{.}\)

Find the \(\threevec{R}{G}{B}\) coordinates for the following colors and check, using the diagrams above, that the two representations agree.
\(\threevec{Y}{C_b}{C_r}=\threevec{128}{0}{0}\text{.}\)
\(\threevec{Y}{C_b}{C_r}=\threevec{128}{60}{0}\text{.}\)
\(\threevec{Y}{C_b}{C_r}=\threevec{128}{0}{60}\text{.}\)

Write an expression for
The luminance \(Y\) as it depends on \(R\text{,}\) \(G\text{,}\) and \(B\text{.}\)
The blue chrominance \(C_b\) as it depends on \(R\text{,}\) \(G\text{,}\) and \(B\text{.}\)
The red chrominance \(C_r\) as it depends on \(R\text{,}\) \(G\text{,}\) and \(B\text{.}\)
Explain how these quantities can be roughly interpreted by stating that
the luminance represents the brightness of the color.
the blue chrominance measures the amount of blue in the color.
the red chrominance measures the amount of red in the color.