TikZ Ellipse and Arc Operations Acting Unexpected

Question

I'm trying to draw the arc of an ellipse which is centered at the origin. I'd like it to start drawing at an initial angle of 45 degrees and end at 90 degrees.

From this question on this site, there is a comment that indicates one way to do it is

\documentclass{article}
\usepackage{tikz}
\usepackage[dvipsnames]{xcolor}

\begin{document}

\begin{tikzpicture}
\draw[fill=ProcessBlue!30] (45:2cm and 1cm) arc [start angle = 45.0, end angle=90.0, x radius = 2cm, y radius = 1cm];
\end{tikzpicture}

\end{document}

which produces

Unless I am being stupid, this doesn't appear to actually be the drawing at the angle of 45 degrees relative to the horizontal. The coordinates of that point are (r*cos(pi/4), r*sin(pi/4)), where r is calculated using this formula from wikipedia. That point is outlined in red below. I also drew a line from the origin to show that that point is calculated correctly.

Here's what I used to make the above.

\documentclass{article}
\usepackage{tikz}
\usepackage[dvipsnames]{xcolor}

\begin{document}
\begin{tikzpicture}
    \draw (-4,0) -- (4,0);
    \draw (0,-4) -- (0,4);
    
    \draw[fill=blue] (45:2cm and 1cm) circle (0.05cm);
    \draw[fill=ProcessBlue!30] (45:2cm and 1cm) arc [start angle = 45.0, end angle=90.0, x radius = 2cm, y radius = 1cm];
    \draw[fill=red] (0.894427190999916, 0.894427190999916) circle (0.05cm);
    \draw[dashed] (0,0) ellipse (2cm and 1cm);
    \draw (0,0) -- (1,1);
\end{tikzpicture}

\end{document}

My question: It appears that the command \draw[fill=ProcessBlue!30] (45:2cm and 1cm) arc... is NOT drawing an arc which starts from the point on the ellipse with axes 2cm and 1cm, that makes a 45 degree angle with the horizontal (Confusingly, that IS how one would do it for a circle...). I think this is the first time I've found TikZ syntax to be so wildly off of what I'd naturally expect. So, what is it actually doing (i.e., why did it start at the blue point)? Is there a simple way to do what I originally aimed to do (I emphasize simple because what I want to do is not very advanced.)?

Answer 1

Regarding my comment...I changed all the 45 to 63.43, which is 90-atan(1/2).

With the 45, you are drawing the arc from the 45 of an unstretched circle. After stretching the circle into an ellipse, the 45 degree angle becomes flattened.

In conclusion, 63.43 deg on the unit circle relative to the x axis becomes 45 degrees after you stretch the x axis by a factor of 2x.

\documentclass{article}
\usepackage{tikz}
\usepackage{xcolor}

\begin{document}
\begin{tikzpicture}
    \draw (-4,0) -- (4,0);
    \draw (0,-4) -- (0,4);
    
    \draw[fill=blue] (63.43:2cm and 1cm) circle (0.05cm);
    \draw[fill=blue!30] (63.43:2cm and 1cm) arc [start angle = 63.43, end angle=90.0, x radius = 2cm, y radius = 1cm];
    \draw[fill=red] (0.894427190999916, 0.894427190999916) circle (0.05cm);
    \draw[dashed] (0,0) ellipse (2cm and 1cm);
    \draw (0,0) -- (1,1);
\end{tikzpicture}

\end{document}

Answer 2

There is no inconsistency. The points are parametrized by (a cos(angle), b sin(angle)).

\documentclass{article}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
    \draw (-4,0) -- (4,0);
    \draw (0,-4) -- (0,4);
    \draw[fill=blue] 
        (45:2cm and 1cm) arc [start angle = 45.0, 
        end angle=90.0, x radius = 2cm, y radius = 1cm];
    \draw[dashed] (0,0) ellipse[x radius=2cm,y radius=1cm];
    \path ({2*cos(45)},{1*sin(45)}) node[circle,inner sep=1pt,fill=red,
        label={[text width=4cm]above right:{I am sitting at 
        $\bigl(a\,\cos(45),b\,\sin(45)\bigr)$ with $a=2$ and $b=1$.}}]{};   
\end{tikzpicture}

\end{document}