The unit ball of quadratic forms on an octagonal sector

Let $$\mathfrak{O}=\{(x,y)\in [0,1{]}^2:x+y\le \sqrt{2}\}$$ be the first-quadrant sector of a regular octagon. For quadratic forms $P(x,y)=a{x}^2+bxy+c{y}^2$, we study the supremum norm over $\mathfrak{O}$. We obtain a complete five-region formula for the norm, according to whether the norming contact occurs at an endpoint or in the interior of one of the three radial sides. We then prove that the projection of the unit ball onto the $ac$-plane is exactly $[-1,1{]}^2$, compute both endpoints of every vertical section, and thereby parametrize the entire unit sphere. Finally, we characterize the extreme points of the unit ball as four explicit curves, their negatives, and four pairs of isolated points. The resulting description is fully explicit and reduces subsequent convex extremal problems on this polynomial space to four one-parameter families and finitely many isolated polynomials.