7.7 Lines and Planes in 3D Space: Glossary
The pronunciation guide given in these glossaries draws on the symbols International Phonetic Alphabet. We have made two adaptations to the IPA system. Firstly, we use slanted lines as opposed to square brackets to enclose pronunciation material. We do this as the slanted line convention is more common in language teaching contexts. Secondly, we use the symbol <r> to denote the voiced alveolar approximant sound which appears in British English at the beginning of the each word in the phrase 'red rhombus'; the IPA would instead use an inverted 'r' symbols namely <ɹ>. We have done this to avoid unnecessary complication to the visuals of the glossaries.
In addition to these conventions, please also note that the pronunciations given are indicative of contemporary Standard British English pronunciation and should not be read as normative, nor as a target pronunciation that you must aim to produce. The pronunciation of English varies widely across World Englishes; part of becoming fluent in the academic English of mathematics is developing an awareness of, and sensitivity to, the variation in pronunciation that you will encounter.
| Word | Pronunciation | Part of speech词性 | Related terminology | Chinese meaning | Example sentence 1 | Example sentence 2 |
|---|---|---|---|---|---|---|
| graph | /ɡrɑːf/ | n. | 图像 (通常指函数、曲线等在坐标系中的图像) | The graph shows the relationship between variables. 图像显示变量关系。 |
We graph the function \(y=x^2\). 我们画函数\(y=x^2\)的图像。 |
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| diagram | /ˈdaɪəɡræm/ | n. | 示意图，几何图 （不一定有坐标轴，主要用于几何，结构关系，流程说明） | We draw a diagram to understand the problem. 我们画图来理解问题。 |
The diagram illustrates the geometric relationship. 图示说明几何关系。 |
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| plot | /plɒt/ | v./n. | 作图，描点；图 （强调画图过程） | Plotting the function reveals its shape. 作图可以显示函数形状。 |
We plot the data points on a coordinate plane. 我们在坐标系中绘制数据点。 |
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| collinear | /kəʊˈlɪniə/ | adj. | non-collinear 不共线的 | 共线的 | Vectors \({\bf u}, {\bf v}\) are collinear iff \({\bf u} \times {\bf v} = 0\). 向量 \({\bf u}, {\bf v}\) 共线当且仅当 \({\bf u} \times {\bf v} = 0\)。 |
If \(P\) lies on line \(AB\), then \(A, P, B\) are collinear. 若点 \(P\) 在直线 \(AB\) 上，则 \(A, P, B\) 三点共线。 |
| co-planar | /ˌkəʊˈpleɪnə(r)/ | adj. | 共面的 | The two lines are co-planar. 这两条直线是共面的。 |
The points \(A, B, C\), and \(D\) are co-planar. 点 \(A, B, C\) 和 \(D\) 是共面的。 |
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| skew lines | /skjuː laɪnz/ | n. phr. | 异面直线 | Skew lines have no common point and different direction vectors. 异面直线无公共交点且方向向量不同。 |
Compute the distance between two skew lines in \(\mathbb{R}^3\). 计算三维空间 \(\mathbb{R}^3\) 中两条异面直线间的距离。 |
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| radius (plural: radii) | /ˈreɪdiəs/, /ˈreɪdiaɪ/ | n. | 半径 | The radius determines the size of the circle. 半径决定圆的大小。 |
The radius of a circle is half the diameter. 半径是直径的一半。 |
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| diameter | /daɪˈæmɪtə(r)/ | n. | 直径 | The diameter passes through the center. 直径经过圆心。 |
The diameter is twice the radius. 直径是半径的两倍。 |
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| circumference | /səˈkʌmfərəns/ | n. | 周长 | We calculate the circumference of the circle. 我们计算圆的周长。 |
The circumference is \(2 \pi r\). 周长是\(2 \pi r\)。 |
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| arc | /ɑːk/ | n. | arclength 弧长 | 弧 | The length of an arc depends on the radius. 弧长取决于半径。 |
An arc is part of a circle. 弧是圆的一部分。 |
| segment | /ˈseɡmənt/ | n. | 线段 | The segment \(AB\) connects two points. 线段\(AB\)连接两个点。 |
A line segment has two endpoints. 线段有两个端点。 |
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| angle | /ˈæŋɡl/ | n. | right angle 直角 | 角 | The angle between vectors is \(60\) degrees. 向量之间的夹角是\(60\)度。 |
An angle is formed by two rays. 角由两条射线形成。 |
| triangle | /ˈtraɪæŋɡl/ | n. | 三角形 | The sum of angles in a triangle is \(180\) degrees. 三角形内角和为\(180\)度。 |
A triangle has three sides and three angles. 三角形有三条边和三个角。 |
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| rectangle | /ˈrektæŋɡl/ | n. | parallelogram 平行四边形； rhombus 菱形； pentagon 五边形； hexagon 六边形 |
矩形 | The area of a rectangle is length times width. 矩形面积为长乘宽。 |
A rectangle has opposite sides equal and parallel. 矩形对边相等且平行。 |
| ellipse | /ɪˈlɪps/ | n. | 椭圆 | The orbit of planets is an ellipse. 行星轨道是椭圆。 |
An ellipse has two focal points. 椭圆有两个焦点。 |
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| parabola | /pəˈræbələ/ | n. | asymptote 渐近线 | 抛物线 | A parabola opens upward when the coefficient is positive. 当系数为正时抛物线开口向上。 |
The graph of a quadratic function is a parabola. 二次函数图像是抛物线。 |
| spiral | /ˈspaɪrəl/ | n. | 螺旋 | Plot the spiral for \(\theta \in [0, 4pi]\) in polar coordinates. 在极坐标下绘制 \(\theta \in [0, 4 \pi]\) 范围内的螺线。 |
The point moves along a spiral toward the origin. 该点沿螺线向原点移动。 |
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| disc | /dɪsk/ | n. | 圆盘，圆形区域 | The closed disc centered at origin is \(D = \{(x,y) \mid x^2 + y^2 \leq1\}\). 以原点为圆心的闭圆盘为 \(D = \{(x,y) \mid x^2 + y^2 \leq 1\}\)。 |
Integrate the function over the disc of radius 2. 在半径为2的圆盘区域上对该函数积分。 |
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| wedge | /wɛdʒ/ | n. | 楔形 | This wedge region lies between two intersecting planes. 该楔形区域位于两个相交平面之间。 |
Project the \(3D\) wedge onto the \(xy\)-plane for integration. 将三维楔体投影到 \(xy\) 平面上以便积分计算。 |
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| sphere | /sfɪə(r)/ | n. | 球体 | The volume of a sphere is \(\frac{4}{3} \pi r^3\). 球体体积是\(\frac{4}{3} \pi r^3\)。 |
A sphere has all points equidistant from center. 球面所有点到中心距离相等。 |
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| cylinder | /ˈsɪlɪndə(r)/ | n. | cylindrical 圆柱形的 | 圆柱 | The volume of a cylinder is \(\pi r^2 h\). 圆柱体积是\(\pi r^2 h\)。 |
A cylinder has circular bases. 圆柱有圆形底面。 |
| cone | /kəʊn/ | n. | 圆锥 | The volume of a cone is one-third of a cylinder. 圆锥体积是圆柱的三分之一。 |
A cone has a circular base and a vertex. 圆锥有一个圆底和一个顶点。 |
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| solid of revolution | /ˈsɒlɪd əv ˌrɛvəˈluːʃən/ | n. phr. | 旋转体 | The region is rotated about the \(x\)-axis to form a solid of revolution. 该区域绕 \(x\) 轴旋转形成一个旋转体。 |
Find the volume of the solid of revolution. 求该旋转体的体积。 |
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| volume | /ˈvɒljuːm/ | n. | 体积 | The volume of a cube is \(a^3\). 立方体体积为\(a^3\)。 |
Volume increases as the cube of the side length. 体积随边长的三次方增长。 |
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| contour | /ˈkɒntɔː(r)/ | n. | 等高线 | A contour plot shows level curves. 等高线图显示水平曲线。 |
Contour lines represent constant values. 等值线表示常数值。 |
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| cusp | /kʌsp/ | n. | 尖点 | A cusp indicates non-smooth behavior. 尖点表示不光滑行为。 |
The graph has a cusp at \(x=0\). 图像在\(x=0\)处有尖点。 |
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| branch | /brɑːntʃ/ | n. | 分支 | Each branch behaves differently. 每个分支行为不同。 |
The function has two branches. 该函数有两个分支。 |
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| strip | /strɪp/ | n. | 带状区域 | The solution lies in a narrow strip on the graph. 解位于图像的窄带中。 |
A strip of numbers satisfies the inequality. 数值带满足该不等式。 |
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| portion | /ˈpɔːʃən/ | n. | 部分，片段 | Integrate over the portion of the sphere where \(z \geq 0\). 在球面满足 \(z \geq 0\) 的部分区域上积分。 |
Solve only the portion of the domain with \(x \gt 0\). 仅求解定义域中 \(x \gt 0\) 的部分。 |
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| curve | /kɜːv/ | n. | 曲线 | A smooth curve has no sharp corners. 光滑曲线没有尖角。 |
The curve represents a function. 曲线表示一个函数。 |
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| surface | /ˈsɜːfɪs/ | n. | 曲面 | The surface area depends on shape. 表面积取决于形状。 |
A surface is a \(3D\) extension of a curve. 曲面是曲线的三维扩展。 |
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| reference point | /ˈrɛfərəns pɔɪnt/ | n. | 参考点 | Take \(P_0=(1,2)\) as the reference point for the line equation. 取点 \(P_0=(1,2)\) 作为直线方程的参考点。 |
Express all vectors relative to the fixed reference point. 以固定参考点为基准表示所有向量。 |
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| symmetric representation | /sɪˈmɛtrɪk ˌrɛprɪzɛnˈteɪʃən/ | n. phr. | symmetric form 对称形式 | 对称表示（对称方程） | Convert parametric equations to symmetric representation directly. 可直接将参数方程转化为对称式方程。 |
The symmetric representation of a line: \(\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}\). 直线的对称式方程：\(\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}\)。 |
| two-dimensional | /ˌtuː daɪˈmɛnʃənəl/ | adj. | three-dimenional 三维的 | 二维的 | \(\mathbb{R}^2\) is the standard two-dimensional Euclidean space. \(\mathbb{R}^2\) 是标准的二维欧几里得空间。 |
A two-dimensional vector has the form \((x,y)\). 二维向量的形式为 \((x,y)\)。 |
