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统计代写|数据可视化代写Data visualization代考|Three-Dimensional Plots

Contour maps and contour plots were certainly useful, but they were still images on a two-dimensional surface, using shading or level curves to show a third dimension. There is a huge difference between trying to navigate a driving route or a hike from a 2D map that shows elevation with isolines versus a 3D relief map that shows elevation in context, using perspective, realistic lighting (“raytracing”), color (“terrain colors”), texture mapping, and other techniques to generate beautiful and more useful 3D topographic maps. ${ }^{10}$

The technique of rendering 3D views in depth and perspective on a flat surface was known to artists for centuries, but early landscapes lacked realism. The first exemplar to get perspective approximately right was the painting View of the Arno Valley by Leonardo da Vinci in 1473, his first known drawing; but that is just an artist’s view. For data graphics, the precise technical details of drawing a 3D surface of a response variable $z$ over a plane defined by $(x, y)$ coordinates did not develop until the late 1800 s. By 1869, in the course of work on thermodynamics, the German physicist Gustav Zeuner [1828-1907] worked out the mathematics of what has come to be called the axonometric projection: a way of drawing a 3D coordinate system so that the coordinate axes looked to be at right angles, and parallel slices or curves had the proper appearance. Zeuner took Descartes to 3D.

An example is shown in Figure 8.6. The coordinate axes, $X, Y, Z$ are shown with the origin in the back. Two parallel curves are drawn, and the goal of this diagram is to explain how the rectangular region can be seen in terms of its projected shadows (shaded) as rectangles on the bottom and left planes.
The first known use of a 3D data graphic using these ideas was designed by Luigi Perozzo [1856-1916], an Italian mathematician, statistician, and,ultimately, a hero of demography, largely for this contribution to the study of the distribution of age over time.

A graphic innovation on this topic appeared in the U.S. Census atlas of 1870, where Francis Walker pioneered the idea of an “age-sex pyramid” showing the age distribution of the population by sex. It was called a pyramid because it compared the populations of men and women in back-to-back histograms by age, in a way that resembled a pyramid. In a number of plates, these data were broken down by state and other factors, in such a way that insurance agencies could begin to set age-, sex-, and region-specific rates for an annuity or life insurance policy. To demographers, this method gave a way to characterize fertility, life expectancy, and other questions regarding population variation. But these were still 2D graphs.

统计代写|数据可视化代写Data visualization代考|Contour Maps

Maps start with a two-dimensional surface defined by latitude and longitude. After geographic features such as rivers, cities, and towns had been inscribed, it was natural for cartographers to want to show features of elevation, and landforms such as mountains and plateaus, in what came to be called topographic maps. This idea was a natural initial impetus for 3D thinking and visual depiction.

The first large-scale topographic map of an entire country was the Carte géométrique de la France, by the French astronomer and surveyor CésarFrançis Cassini de Thury [1714-1784], ${ }^2$ completed in 1789. But well before these precise determinations of altitude were made, map makers began to try to show topographical features using contour lines of equal elevation on their maps. These were useful for finding the way through a mountain range as well as for military defense.

Beyond wayfinding and route navigation, thematic maps use the features of geography to show something more: how some quantity of interest varies from place to place. Figure $3.3$ by Balbi and Guerry is a nice example of the use of shaded (choropleth) maps of France to display the geographic distribution of crimes and compare this with the distribution of literacy. But this and similar maps treat geographic regions as discrete, and simply shade the entire arca in rclation to a variable of interest.

The language and symbolism of maps expanded to display more abstract quantitative phenomena that varied systematically over geographical space. This was technically a small step from topographic maps that showed elevation of terrain using either color shading or iso-curves (lines of equal magnitude), but the impact was profound in scientific investigation. It was essentially what Galton had done in mapping the contours of equal barometric pressure across Europe (see Plate 12).

This idea, of drawing level curves or contours on a map to show a data variable, began much earlier. Perhaps the first complete example ${ }^3$ is the 1701 map by Edmund Halley, showing lines of equal magnetic declination (isogons) for the world, shown here in Figure 8.2. It was titled, in a style that tried to tell the whole story on the frontispiece, The Description and Uses of a New, and Correct Sea-Chart of the Whole World, Shewing Variations of the Compass.

统计代写|数据可视化代写Data visualization代考|STAT1100

统计代写|数据可视化代写数据可视化代考|三维图

.


等高线图和等高线图当然有用,但它们仍然是二维表面上的图像,使用阴影或水平曲线来显示第三维。在驾驶路线或徒步旅行中,使用等高线显示高程的2D地图与使用上下文显示高程的3D浮雕地图之间存在着巨大的区别,3D浮雕地图使用透视、逼真的照明(“光线追踪”)、颜色(“地形颜色”)、纹理映射和其他技术生成漂亮且更有用的3D地形图。${ }^{10}$


几个世纪以来,艺术家们就已经知道在平面上以深度和透视的方式渲染3D视图的技术,但早期的景观缺乏现实感。第一个将透视法近似正确的例子是列奥纳多·达·芬奇(Leonardo da Vinci)在1473年创作的《阿诺谷风景》(View of The Arno Valley),这是他的第一幅已知画作;但这只是艺术家的观点。对于数据图形来说,在$(x, y)$坐标定义的平面上绘制响应变量$z$的三维曲面的精确技术细节直到19世纪后期才发展起来。到1869年,在研究热力学的过程中,德国物理学家古斯塔夫·齐纳(Gustav Zeuner, 1828-1907)提出了后来被称为轴测投影的数学方法:一种绘制三维坐标系的方法,使坐标轴看起来呈直角,平行切片或曲线具有适当的外观。Zeuner把笛卡尔带到了3D世界

示例如图8.6所示。坐标$X, Y, Z$显示在后面,原点在后面。绘制了两条平行曲线,该图的目的是解释如何通过其投影阴影(阴影)作为底部和左侧平面上的矩形来看到矩形区域。已知的第一个使用这些思想的3D数据图表是由意大利数学家、统计学家、最终成为人口学英雄的路易吉·佩罗佐(Luigi Perozzo, 1856-1916)设计的,主要是因为他对年龄随时间分布研究的贡献 1870年的美国人口普查地图集在这一主题上出现了一个图形创新,弗朗西斯·沃克(Francis Walker)率先提出了“年龄-性别金字塔”的想法,按性别显示人口的年龄分布。它被称为金字塔,因为它以一种类似金字塔的方式,将男性和女性的人口按年龄进行了背靠背的直方图比较。在一些车牌上,这些数据被按州和其他因素分解,这样保险机构就可以开始为年金或人寿保险单制定年龄、性别和地区特定的费率。对人口学家来说,这种方法提供了一种描述生育率、预期寿命和其他有关人口变化的问题的方法。但这些仍然是2D图。

统计代写|数据可视化代写数据可视化代考|等高线地图


地图从由经纬度定义的二维表面开始。在河流、城市和城镇等地理特征被记录下来之后,制图者自然想要在后来被称为地形图的东西中显示海拔和地形的特征,如山脉和高原。这个想法是3D思维和视觉描绘的自然初始动力

第一张关于整个国家的大比例尺地形图是由法国天文学家和测量员CésarFrançis卡西尼·德·图里[1714-1784]制作的《法国地图géométrique de la France》,${ }^2$完成于1789年。但早在精确确定海拔高度之前,地图绘制者就开始尝试在地图上使用等高线来显示地形特征。这些在寻找穿过山脉的路和军事防御方面都很有用


除了寻路和路线导航之外,专题地图还利用地理特征来显示更多的东西:兴趣的数量如何因地而异。Balbi和Guerry的图$3.3$是使用法国阴影(choropleth)地图来显示犯罪的地理分布并将其与识字率分布进行比较的一个很好的例子。但这幅地图和类似的地图将地理区域视为离散的,并简单地将循环中的整个弧区遮蔽为一个感兴趣的变量


地图的语言和象征符号扩展到显示更抽象的数量现象,这些现象在地理空间中有系统地变化。从技术上讲,这与使用彩色阴影或等距曲线(等距线)显示地形高度的地形图相比,只是一小步,但它对科学调查产生了深远的影响。这基本上就是高尔顿在绘制欧洲平均气压等高线时所做的工作(见图12)


在地图上绘制水平曲线或等高线来显示数据变量的想法,在更早的时候就开始了。也许第一个完整的例子${ }^3$是埃德蒙·哈雷1701年的地图,显示了世界的等赤纬线(等角线),如图8.2所示。这本书的标题是《一幅新的、正确的、显示罗盘变化的世界海图的描述和用途》,力图在扉页上说明整个故事

统计代写|数据可视化代写Data visualization代考

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