how to draw 3d graphs on paper
Suppose you want to plot $z = f(x, y)$ over the rectangle $[a, b] \times [c, d]$, i.due east., for $a \leq x \leq b$ and $c \leq y \leq d$, using a mesh filigree of size $yard \times due north$. One uncomplicated arroyo is to use "orthogonal projection":
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Select a function and the rectangle over which you lot want to plot. Find or estimate the minimum and maximum values the role achieves.
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Tack down a sheet of paper on a drafting board. Using a T-square, $thirty$-$60$-$90$ triangle, and ruler, lay out a parallelogram on your paper with ane side horizontal, the rectangular domain seen in perspective, and mark off the subdivision points along the outer edges ($1000$ equal intervals in the $ten$-direction and $n$ equal intervals in the $y$-direction). Use the minimum and maximum values of the office to estimate where on the paper the domain should be drawn, and to make up one's mind on the overall vertical calibration of the plot.
For definiteness (run across diagram below), let'south call the bottom edge of the parallelgram $x = x_{0}$ and the left border $y = y_{0}$. (Depending on how the parallelogram is oriented, you might take $x_{0} = a$ or $b$, and $y_{0} = c$ or $d$.) Using a sharp 6H pencil, subdivide the parallelogram (the domain) into an $m \times northward$ grid.
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Calculate the step sizes $$ \Delta ten = \frac{b - a}{m},\qquad \Delta y = \frac{d - c}{due north}. $$ (The formulas beneath assume the footstep sizes are positive, i.e., thet $x_{0} = a$ and $y_{0} = c$. The modifications should be fairly obvious if $10$ decreases from bottom to superlative and/or $y$ decreases from left to right.)
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To effect hidden line removal, we'll plot front to back. Calculate the "front row" values $$ f(x_{0}, y_{0} + j\, \Delta y),\qquad 1 \leq j \leq n. $$ Locate each point $(x_{0}, y_{0} + j\, \Delta y)$ in your grid, measure out up or downward to the advisable height, and put a dot at that location. When you lot're plotted these $due north$ points, connect the dots from left to right with a 2B pencil.
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Now iterate the post-obit step, letting $i$ run from $one$ to $g$. Calculate the values $$ f(x_{0} + i\, \Delta x, y_{0} + j\, \Delta y),\qquad 1 \leq j \leq n. $$ Locate each point $(x_{0} + i\, \Delta x, y_{0} + j\, \Delta y)$ in your grid, and mensurate upwards or downwardly to the appropriate height. When you're plotted these $due north$ points:
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Draw 1 row of "front-to-back" segments: For each $j = 1, \dots, n$, connect the dot over $(x_{0} + (i - 1)\, \Delta ten, y_{0} + j\, \Delta y)$ to the dot over $(x_{0} + i\, \Delta x, y_{0} + j\, \Delta y)$. (Use calorie-free lines or no lines if the segment lies behind a office of the surface yous have already plotted.)
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Draw the $i$th row: For each $j = i, \dots, northward$, connect the dot over $(x_{0} + i\, \Delta ten, y_{0} + j\, \Delta y)$ to the dot over $(x_{0} + i\, \Delta x, y_{0} + (j + one)\, \Delta y)$. (Once more, use light lines or no lines if the segment lies backside a part of the surface y'all accept already plotted.)
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Speaking from experience, the process takes (with a calculator) about eight hours for a $20 \times 20$ mesh. It's doubtless faster to tabulate all the values of $f$ at the mesh points, then to plot points by reading from the table. (I was impetuous every bit a student, and alternately calculated one value and plotted ane point.)
The diagram shows (a computer-fatigued version of) the get-go role I plotted, shifted up to avoid overlap with the rectangular mesh in the domain. When you lot're actually plotting on paper, you lot probably don't want to waste the vertical space, and so volition have to draw the grid lightly and plot over it.
Source: https://math.stackexchange.com/questions/2257410/is-there-a-way-to-draw-3d-graphs-on-paper
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