#set document(title: "8.3 3D Plots", author: "OpenStax") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 8.3#h(0.6em)3D Plots #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Prerequisites] Introduction to Bivariate Data #linebreak() #linebreak() ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Learning Objectives] + Describe a 3D Plot. + Give an example of the value of a 3D plot. ] Just as two-dimensional scatter plots show the data in two dimensions, 3D plots show data in three dimensions. Figure 1 shows a 3D scatter plot of the fat, non-sugar carbohydrates, and calories from a variety of cereal types. #figure(figph[Three-dimensional scatterplot of breakfast cereals with axes Carbohydrates (10 to 55), Fat (0 to 10) and Calories (50 to 250), drawn as a wireframe box with dropped gridlines to each point. Most cereals cluster at low fat with carbohydrates between about 20 and 30 and calories between about 100 and 150; a few high-calorie points sit near the top of the box.], alt: "Three-dimensional scatterplot of breakfast cereals with axes Carbohydrates (10 to 55), Fat (0 to 10) and Calories (50 to 250), drawn as a wireframe box with dropped gridlines to each point. Most cereals cluster at low fat with carbohydrates between about 20 and 30 and calories between about 100 and 150; a few high-calorie points sit near the top of the box.", caption: [Figure 1. A 3D scatter plot showing fat, non-sugar carbohydrates, and calories from a variety of cereal types.]) Many statistical packages allow you to rotate the axes interactively to view the data from a different vantage point. Figure 2 is an example. #figure(figph[The same three cereal variables plotted in three dimensions but with the box rotated to a different vantage point, so Fat now runs up the left-hand axis and Calories along the top. The cluster of low-fat cereals now reads as a dense flat sheet near the bottom, and the spread in calories is easier to see than in the first orientation.], alt: "The same three cereal variables plotted in three dimensions but with the box rotated to a different vantage point, so Fat now runs up the left-hand axis and Calories along the top. The cluster of low-fat cereals now reads as a dense flat sheet near the bottom, and the spread in calories is easier to see than in the first orientation.", caption: [Figure 2. An alternative 3D scatter plot showing fat, non-sugar carbohydrates, and calories.]) A fourth dimension can be represented as long as it is represented as a nominal variable. Figure 3 represents the different manufacturers by using different colors. #figure(figph[The same rotated cereal plot with a fourth, nominal variable added: each point is colored by manufacturer (red, blue, green and orange). The colors are interleaved through the main low-fat cluster rather than separating into distinct blocks.], alt: "The same rotated cereal plot with a fourth, nominal variable added: each point is colored by manufacturer (red, blue, green and orange). The colors are interleaved through the main low-fat cluster rather than separating into distinct blocks.", caption: [Figure 3. The different manufacturers are color coded.]) Interactively rotating 3D plots can sometimes reveal aspects of the data not otherwise apparent. Figure 4 shows data from a pseudo random number generator. Figure 4 does not show anything systematic and the random number generator appears to generate data with properties similar to those of true random numbers. #figure(figph[Three-dimensional scatterplot of 400 X, Y, Z triples from a pseudo random number generator, shown inside a wireframe cube. From this vantage point the points look evenly and formlessly scattered throughout the cube — nothing systematic is visible, so the generator appears to behave like a true random source.], alt: "Three-dimensional scatterplot of 400 X, Y, Z triples from a pseudo random number generator, shown inside a wireframe cube. From this vantage point the points look evenly and formlessly scattered throughout the cube — nothing systematic is visible, so the generator appears to behave like a true random source.", caption: [Figure 4. A 3D scatter plot showing 400 values of X, Y, and Z from a pseudo random number generator.]) Figure 5 shows a different perspective on these data. Clearly they were not generated by a random process. #figure(figph[The same 400 pseudo-random triples viewed from a different angle. The points are no longer formless: they lie on a series of evenly spaced parallel diagonal planes running across the cube, with empty space between them. Rotating the plot exposes the lattice structure that the first view concealed — these values were not generated by a random process.], alt: "The same 400 pseudo-random triples viewed from a different angle. The points are no longer formless: they lie on a series of evenly spaced parallel diagonal planes running across the cube, with empty space between them. Rotating the plot exposes the lattice structure that the first view concealed — these values were not generated by a random process.", caption: [Figure 5. A different perspective on the 3D scatter plot showing 400 values of X, Y, and Z from a pseudo random number generator. #linebreak() #linebreak()]) Figures 4 and 5 are reproduced with permission from #link("http://rsnippets.blogspot.com/2011/11/plotting-randu-dataset.html")[R snippets] by Bogumil Kaminski.