Level curves of the objective

Parametric equations

The objective function

has level lines $px+qy=k$, a family of parallel lines with a fixed slope. The normal vector $\mathbf{n}=(p,q)$ points where $z$ increases fastest. Moving a level line along $\mathbf{n}$ makes $k$ increase or decrease monotonically.

This family splits the whole plane into layers. Every point lies on exactly one level line, and that line’s $k$ is the point’s $z$ value.

Interaction

• Coefficient p / Coefficient q: change the two coefficients of the objective. The slope of the family changes, but the lines in it always stay parallel
• Level k: move the current level line. You can also drag the thick line in the figure, and $k$ follows the $z$ value at the pointer
• Level line family: show or hide several neighboring level lines. The “Spacing” reading in the sidebar is $\Delta k/\Vert \mathbf{n}\Vert$. The larger $\Vert \mathbf{n}\Vert$ is, the closer the same difference in $k$ packs the lines
• Test point: drag the blue point in the figure, read the $k$ of the level line through it, and check it against $z=px+qy$
• The white arrow is the normal $\mathbf{n}=(p,q)$. It points where $z$ increases fastest, and the level lines are always perpendicular to it

What to notice

• The level lines are parallel to each other. A different $k$ moves the line, but the slope stays the same.
• The normal vector $\mathbf{n}=(p,q)$ is perpendicular to the level lines, and $z$ changes monotonically along it.
• The objective turns every point of the plane into a number. A level line is the set of points that give the same number.

Related

• Constraint half-planes and the feasible region
• Finding the optimal solution at a vertex
• Linear programming

Further reading

• Linear programming (Wikipedia)
• Contour line (Wikipedia)