Geometry in architecture and building Hans Sterk Faculteit Wiskunde en Informatica Technische Universiteit Eindhoven
Geometry in architecture and building
Apr 05, 2023
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meetkunde.dviii
Lecture notes for ‘2DB60 Meetkunde voor Bouwkunde’
The picture on the cover was kindly provided by W. Huisman, Department of Architecture and Building. The picture shows the Olympic Stadium for the 1972 Olympic Games in Munich, Germany, designed by Frei Otto.
c© 2005–2008 Faculteit Wiskunde en Informatica, TU/e; Hans Sterk
Contents
1 Shapes in architecture 1
1.1 A brief tour of shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Different perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Numbers in architecture: The golden ratio . . . . . . . . . . . . . . . . . . 11 1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 3–Space: lines and planes 21
2.1 3–space and vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Describing lines and planes . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 The relative position of lines and planes: intersections . . . . . . . . . . . . 33 2.4 The relative position of lines and planes: angles . . . . . . . . . . . . . . . 34 2.5 The relative position of points, lines and planes: distances . . . . . . . . . 37 2.6 Geometric operations: translating lines and planes . . . . . . . . . . . . . . 41 2.7 Geometric operations: rotating lines and planes . . . . . . . . . . . . . . . 42 2.8 Geometric operations: reflecting lines and planes . . . . . . . . . . . . . . . 44 2.9 Tesselations of planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 Quadratic curves, quadric surfaces 57
3.1 Plane quadratic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2 Parametrizing quadratic curves . . . . . . . . . . . . . . . . . . . . . . . . 67 3.3 Quadric surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.4 Parametrizing quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.5 Geometric ins and outs on quadrics . . . . . . . . . . . . . . . . . . . . . . 81 3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4 Surfaces 91
4.1 Describing general surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.2 Some constructions of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.3 Surfaces: tangent vectors and tangent planes . . . . . . . . . . . . . . . . . 99 4.4 Surface area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.5 Curvature of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.6 There is much more on surfaces . . . . . . . . . . . . . . . . . . . . . . . . 111
i
5 Rotations and projections 117
5.1 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.2 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.3 Parallel projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Chapter 1
1.1 A brief tour of shapes
1.1.1 Take a look at modern architecture and you will soon realize that the last decades have produced an increasing number of buildings with exotic shapes. Of course, also in earlier times the design of buildings has been influenced by mathematical ideas regarding, for instance, symmetry. Both historical and modern developments show that mathematics can play an important role, ranging from appropriate descriptions of designs to guiding the designer’s intuition. This course aims at providing the mathematical tools to describe various types of shapes in a mathematical way and to manipulate them. In handling them in more involved situations, mathematical computer software such as Maple is very useful. This section discusses a few examples of architectural shapes and hints at the relevant mathematics. Related to these shapes you can think of questions like the following. How can I describe this object with equations? How can I down-size the object, or make it more curved? How does the surface area or volume change if the designer changes the position of a wall?
1.1.2 Some words on coordinates
Geometry deals with shapes, but in actually handling these shapes, it is profitable to bring them within the mathematical realm of numbers and equations. The usual way to get numbers in relation to shapes in your hands is through the use of coordinates. There are many coordinate systems, but the most common coordinate system is the familiar cartesian coordinate system, where you choose an origin in 3–space and three mutually perpendicular axes through the origin (often, but not necessarily, labelled as x–axis, y– axis and z–axis), etc. Each point in space is then characterized by its three coordinates, for instance (2,−
√ 3, 0). (In 2–space, only two axes are needed and points are described
by a pair of coordinates.) We usually refer to coordinatized 2–space and 3–space as R 2 and
R 3, respectively. Equations, like x + 2y + 3z = −5, describe shapes in 3–space: A point
(x, y, z) in 3–space is on the plane precisely if its coordinates satisfy the equation. In this case the resulting shape is a plane. All sorts of geometric operations have their algebraic counterparts. For example, the result of reflecting the point P = (x, y, z) in the x, y–
1
2 Shapes in architecture
plane is the point with coordinates (x, y,−z). Rotating the point around the z–axis over 90 yields the point (−y, x, z) or (y,−x, z), depending on the orientation of the rotation. Coordinates of some sort and the corresponding algebraic machinery are at the basis of computations and of useful implementations in computer software. This mix of shapes and numbers is central in this course.
Of course, it is up to the user to choose a convenient origin and to fix the direction of the axes. Two designers may have decided to use different coordinate systems. To be able to deal with each other’s data they are confronted with the question how to transform one system of coordinates into the second one. For instance, if you view a building from two different points, then how are the two viewpoints related exactly?
1.1.3 A brief word on lines and planes
Flat objects are easier to describe than curved ones. So in Chapter 2 flat objects like lines and planes will be discussed before we turn to a more detailed study of curved objects in later chapters. Here is a tiny preview.
Figure 1.1: A plane in 3–space, usually described by an equation of the form ax+ by+ cz = d. Of course, only part of the plane is drawn, since the plane extends indefinitely. If we describe walls by planes, we need to be aware of the fact that only part of the plane corresponds to the wall.
Suppose we work with ordinary cartesian coordinates x and y in the plane. A line in the plane is described by an equation in the variables x and y of the form
ax + by = c,
such as 2x − 7y = 9. A plane in space involves a linear equation in one more variable, z:
ax + by + cz = d.
For example, 3x + 2y − 7z = 9 is a plane in 3–space. These equations provide implicit descriptions: you know the condition the coordinates have to satisfy in order to be the coordinates of a point of the line or plane. There are also explicit descriptions for lines and planes, so–called parametric descriptions. This chapter is not the place to discuss these matters in detail. Instead we give a sketch.
1.1 A brief tour of shapes 3
Let us consider the line in the plane with equation 2x + 3y = 6. We can solve this equation for y in terms of x: y = (6 − 2x)/3. If we assign the value λ to x, then the pair can be described as
x = λ y = 2 − 2λ/3.
We rewrite this as
(x, y) = (0, 2) + λ(1,−2/3),
so that the relation with points in R 2 comes out more clearly. Substituting any value
for λ in the expression on the right-hand side (no condition on λ) produces the explicit coordinates of a point on the line. For instance, for λ = 9, the corresponding point on the line is (9, 2 − 9 · 2/3) = (9,−4). The parametric description (x, y) = (0, 2) + λ(1,−2/3) also has a clear geometric interpretation: draw a line through (0, 2) whose slope is −2/3.
There are more ways of writing down the solutions of the equation 2x+3y = 6 explicitly. For instance,
(x, y) = (3, 0) + µ(3,−2)
describes the same line! (In fact, we have found this parametric description by solving x in terms of y.) To check that these points are on the line, just plug the corresponding values of x and y into the equation, i.e., substitute x = 3 + 3µ and y = −2µ into 2x + 3y and verify that the resulting expression simplifies to 6:
2(3 + 3µ) + 3(−2µ) = 6 + 6µ − 6µ = 6.
This last representation has the slight advantage, at least for humans — computers don’t mind that much, that there are no fractions in the expression. Again, the parametric description (x, y) = (3, 0) + µ(3,−2) is easy to represent graphically: just start at the point (3, 0) and then draw the line through (3, 0) with slope −2/3 (or: for every 3 steps to the right go 2 steps down).
(3,0)
y
x
2 down
Figure 1.2: The graphical representation of (x, y) = (3, 0) + µ(3,−2).
With fairly elementary techniques you can switch from parametric descriptions to equa- tions and vice versa. For instance, starting with (x, y) = (3, 0)+µ(3,−2), you first extract
4 Shapes in architecture
x = 3 + 3µ and y = −2µ, then add two times the first equality and three times the second one to find that x and y satisfy
2x + 3y = 2(3 + 3µ) + 3 · (−2µ) = 6
(the addition was set up in order to make µ drop out), i.e., 2x + 3y = 6. More aspects of lines and equations will be dealt with in the exercises and in the
following chapters.
1.1.4 Buildings with flat walls
Here begins our trip along various architectural objects. Take a look at the picture of the Van Abbe museum in Eindhoven (Fig 1.3)1.
Figure 1.3: The Van Abbemuseum in Eindhoven (Photo: Peter Cox)
The extension was designed by the Dutch architect Abel Cahen. The walls of the museum are flat or planar, but some of them are sloping walls. Obvious questions are: How much are they inclined? Where do two walls meet exactly? What angle subtend two of these planes? What would change if you change such an angle a bit (just think of the surface area, the position of the roof, etc.)? Of course, mathematics is intended here to support the designing process of the architect. It is no substitute for the architect’s creativity. Let us take a closer look at two of these questions: the intersection of two walls and the angle between two walls.
a1x + b1y + c1z = d1, a2x + b2y + c2z = d2,
(where a1, a2, etc., are the coefficients of the equations).
1http://www.vanabbemuseum.nl/nederlands/gebouw/
Intersecting two planes
The two walls meet along a line, but which one? And how does this line change if the architect decides to change one or both of the planes in the design? Here is a concrete example of dealing with the intersection of two planes (but this chapter is not the place to discuss the techniques used in detail):
x + y + z = 3 2x + y − z = 5
We manipulate the equations in such a way that both x and y can be expressed in terms of z. To do so, we need two steps:
a) We first try to eliminate x from the second equation. By subtracting the first equation
Figure 1.4: Sloping walls.
x + y + z = 3 −y − 3z = −1
b) Next, we add the new second equation to the first one in order to get rid of the variable y in the first equation. We find
x − 2z = 2 −y − 3z = −1
Now, both x and y can be expressed in terms of z as follows: x = 2 + 2z and y = 1 − 3z. Introduce a parameter λ by z = λ. Then we get:
x = 2 + 2λ y = 1 − 3λ z = λ.
Separating the ‘constant part’ and the ‘variable part’, we usually rewrite this as
(x, y, z) = (2, 1, 0) + λ(2,−3, 1).
6 Shapes in architecture
This notation suggests clearly that, not surprisingly, we are dealing with a line: start at the point (2, 1, 0) and move from there in the direction of (2,−3, 1) by varying λ.
The angle between two planes
From the equations x + y + z = 3 and 2x + y − z = 5, the angle φ between the planes can be computed. The relevant information is contained in the coefficients of x, y and z of both equations (the coefficients 3 and 5 on the right–hand side are irrelevant). The three coefficients of the first equation lead to (1, 1, 1). It turns out that the direction from (0, 0, 0) to (1, 1, 1) is perpendicular to the first plane (more on this in Chapter 2). Likewise, the three coefficients of the second equation lead to (2, 1,−1), and the direction from (0, 0, 0) to (2, 1,−1) is perpendicular to the second plane. In this setting, where directions come into play, we usually speak of vectors. The angle between the two planes equals the angle between these two ‘vectors’ (make a picture to convince yourselves). It turns out (and this will be dealt with more extensively in Chapter 2) that the cosine of the angle is computed as follows from the two vectors (1, 1, 1) and (2, 1,−1):
cos φ = 1 · 2 + 1 · 1 + 1 · (−1)√
12 + 12 + 12 · √
22 + 12 + (−1)2 =
√ 6
= 2
3 √
2 =
√ 2
3 .
From this expression, the angle is easily found to be approximately 1.08 radians or 61.9 degrees.
Varying the plane
If we replace one or more of the coefficients in the equation of one of the planes by an (expression in an) auxiliary parameter, we obtain a varying family of planes : for each value of the parameter a plane is defined. Such a family may be useful in the design of a building where you have to specify the position of a wall, say, given that it passes through certain points, intersects the ground level along a certain line, etc. Here are a few examples.
For every value of a the equation x + y + z = a describes a plane. All these planes are parallel to one another (they are all perpendicular to the vector (1, 1, 1)). The plane with equation x+y+z = 0 (i.e., a = 0) contains the origin (0, 0, 0), but the plane with equation x + y + z = 1 evidently does not. You might look for a plane in this family which touches a sphere with center in the origin and given radius.
Here is a family with other properties. The family of planes x + y + az = 3 all pass through the point (3, 0, 0), but no two of them are parallel. They all have the line of intersection of the two planes z = 0 and x + y = 3 in common. Among these planes you might be looking for one which makes an angle of 60 with a horizontal plane.
Rotations and translations are also important ways of varying a plane; these operations will be discussed in later chapters.
1.1.5 Buildings with curved exteriors
Modern buildings show a variety of curved shapes, like the Gherkin in London, see Fig. (1.5)2. To handle these, nonlinear equations and nonlinear parametric descriptions…
Lecture notes for ‘2DB60 Meetkunde voor Bouwkunde’
The picture on the cover was kindly provided by W. Huisman, Department of Architecture and Building. The picture shows the Olympic Stadium for the 1972 Olympic Games in Munich, Germany, designed by Frei Otto.
c© 2005–2008 Faculteit Wiskunde en Informatica, TU/e; Hans Sterk
Contents
1 Shapes in architecture 1
1.1 A brief tour of shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Different perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Numbers in architecture: The golden ratio . . . . . . . . . . . . . . . . . . 11 1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 3–Space: lines and planes 21
2.1 3–space and vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Describing lines and planes . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 The relative position of lines and planes: intersections . . . . . . . . . . . . 33 2.4 The relative position of lines and planes: angles . . . . . . . . . . . . . . . 34 2.5 The relative position of points, lines and planes: distances . . . . . . . . . 37 2.6 Geometric operations: translating lines and planes . . . . . . . . . . . . . . 41 2.7 Geometric operations: rotating lines and planes . . . . . . . . . . . . . . . 42 2.8 Geometric operations: reflecting lines and planes . . . . . . . . . . . . . . . 44 2.9 Tesselations of planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 Quadratic curves, quadric surfaces 57
3.1 Plane quadratic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2 Parametrizing quadratic curves . . . . . . . . . . . . . . . . . . . . . . . . 67 3.3 Quadric surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.4 Parametrizing quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.5 Geometric ins and outs on quadrics . . . . . . . . . . . . . . . . . . . . . . 81 3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4 Surfaces 91
4.1 Describing general surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.2 Some constructions of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.3 Surfaces: tangent vectors and tangent planes . . . . . . . . . . . . . . . . . 99 4.4 Surface area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.5 Curvature of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.6 There is much more on surfaces . . . . . . . . . . . . . . . . . . . . . . . . 111
i
5 Rotations and projections 117
5.1 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.2 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.3 Parallel projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Chapter 1
1.1 A brief tour of shapes
1.1.1 Take a look at modern architecture and you will soon realize that the last decades have produced an increasing number of buildings with exotic shapes. Of course, also in earlier times the design of buildings has been influenced by mathematical ideas regarding, for instance, symmetry. Both historical and modern developments show that mathematics can play an important role, ranging from appropriate descriptions of designs to guiding the designer’s intuition. This course aims at providing the mathematical tools to describe various types of shapes in a mathematical way and to manipulate them. In handling them in more involved situations, mathematical computer software such as Maple is very useful. This section discusses a few examples of architectural shapes and hints at the relevant mathematics. Related to these shapes you can think of questions like the following. How can I describe this object with equations? How can I down-size the object, or make it more curved? How does the surface area or volume change if the designer changes the position of a wall?
1.1.2 Some words on coordinates
Geometry deals with shapes, but in actually handling these shapes, it is profitable to bring them within the mathematical realm of numbers and equations. The usual way to get numbers in relation to shapes in your hands is through the use of coordinates. There are many coordinate systems, but the most common coordinate system is the familiar cartesian coordinate system, where you choose an origin in 3–space and three mutually perpendicular axes through the origin (often, but not necessarily, labelled as x–axis, y– axis and z–axis), etc. Each point in space is then characterized by its three coordinates, for instance (2,−
√ 3, 0). (In 2–space, only two axes are needed and points are described
by a pair of coordinates.) We usually refer to coordinatized 2–space and 3–space as R 2 and
R 3, respectively. Equations, like x + 2y + 3z = −5, describe shapes in 3–space: A point
(x, y, z) in 3–space is on the plane precisely if its coordinates satisfy the equation. In this case the resulting shape is a plane. All sorts of geometric operations have their algebraic counterparts. For example, the result of reflecting the point P = (x, y, z) in the x, y–
1
2 Shapes in architecture
plane is the point with coordinates (x, y,−z). Rotating the point around the z–axis over 90 yields the point (−y, x, z) or (y,−x, z), depending on the orientation of the rotation. Coordinates of some sort and the corresponding algebraic machinery are at the basis of computations and of useful implementations in computer software. This mix of shapes and numbers is central in this course.
Of course, it is up to the user to choose a convenient origin and to fix the direction of the axes. Two designers may have decided to use different coordinate systems. To be able to deal with each other’s data they are confronted with the question how to transform one system of coordinates into the second one. For instance, if you view a building from two different points, then how are the two viewpoints related exactly?
1.1.3 A brief word on lines and planes
Flat objects are easier to describe than curved ones. So in Chapter 2 flat objects like lines and planes will be discussed before we turn to a more detailed study of curved objects in later chapters. Here is a tiny preview.
Figure 1.1: A plane in 3–space, usually described by an equation of the form ax+ by+ cz = d. Of course, only part of the plane is drawn, since the plane extends indefinitely. If we describe walls by planes, we need to be aware of the fact that only part of the plane corresponds to the wall.
Suppose we work with ordinary cartesian coordinates x and y in the plane. A line in the plane is described by an equation in the variables x and y of the form
ax + by = c,
such as 2x − 7y = 9. A plane in space involves a linear equation in one more variable, z:
ax + by + cz = d.
For example, 3x + 2y − 7z = 9 is a plane in 3–space. These equations provide implicit descriptions: you know the condition the coordinates have to satisfy in order to be the coordinates of a point of the line or plane. There are also explicit descriptions for lines and planes, so–called parametric descriptions. This chapter is not the place to discuss these matters in detail. Instead we give a sketch.
1.1 A brief tour of shapes 3
Let us consider the line in the plane with equation 2x + 3y = 6. We can solve this equation for y in terms of x: y = (6 − 2x)/3. If we assign the value λ to x, then the pair can be described as
x = λ y = 2 − 2λ/3.
We rewrite this as
(x, y) = (0, 2) + λ(1,−2/3),
so that the relation with points in R 2 comes out more clearly. Substituting any value
for λ in the expression on the right-hand side (no condition on λ) produces the explicit coordinates of a point on the line. For instance, for λ = 9, the corresponding point on the line is (9, 2 − 9 · 2/3) = (9,−4). The parametric description (x, y) = (0, 2) + λ(1,−2/3) also has a clear geometric interpretation: draw a line through (0, 2) whose slope is −2/3.
There are more ways of writing down the solutions of the equation 2x+3y = 6 explicitly. For instance,
(x, y) = (3, 0) + µ(3,−2)
describes the same line! (In fact, we have found this parametric description by solving x in terms of y.) To check that these points are on the line, just plug the corresponding values of x and y into the equation, i.e., substitute x = 3 + 3µ and y = −2µ into 2x + 3y and verify that the resulting expression simplifies to 6:
2(3 + 3µ) + 3(−2µ) = 6 + 6µ − 6µ = 6.
This last representation has the slight advantage, at least for humans — computers don’t mind that much, that there are no fractions in the expression. Again, the parametric description (x, y) = (3, 0) + µ(3,−2) is easy to represent graphically: just start at the point (3, 0) and then draw the line through (3, 0) with slope −2/3 (or: for every 3 steps to the right go 2 steps down).
(3,0)
y
x
2 down
Figure 1.2: The graphical representation of (x, y) = (3, 0) + µ(3,−2).
With fairly elementary techniques you can switch from parametric descriptions to equa- tions and vice versa. For instance, starting with (x, y) = (3, 0)+µ(3,−2), you first extract
4 Shapes in architecture
x = 3 + 3µ and y = −2µ, then add two times the first equality and three times the second one to find that x and y satisfy
2x + 3y = 2(3 + 3µ) + 3 · (−2µ) = 6
(the addition was set up in order to make µ drop out), i.e., 2x + 3y = 6. More aspects of lines and equations will be dealt with in the exercises and in the
following chapters.
1.1.4 Buildings with flat walls
Here begins our trip along various architectural objects. Take a look at the picture of the Van Abbe museum in Eindhoven (Fig 1.3)1.
Figure 1.3: The Van Abbemuseum in Eindhoven (Photo: Peter Cox)
The extension was designed by the Dutch architect Abel Cahen. The walls of the museum are flat or planar, but some of them are sloping walls. Obvious questions are: How much are they inclined? Where do two walls meet exactly? What angle subtend two of these planes? What would change if you change such an angle a bit (just think of the surface area, the position of the roof, etc.)? Of course, mathematics is intended here to support the designing process of the architect. It is no substitute for the architect’s creativity. Let us take a closer look at two of these questions: the intersection of two walls and the angle between two walls.
a1x + b1y + c1z = d1, a2x + b2y + c2z = d2,
(where a1, a2, etc., are the coefficients of the equations).
1http://www.vanabbemuseum.nl/nederlands/gebouw/
Intersecting two planes
The two walls meet along a line, but which one? And how does this line change if the architect decides to change one or both of the planes in the design? Here is a concrete example of dealing with the intersection of two planes (but this chapter is not the place to discuss the techniques used in detail):
x + y + z = 3 2x + y − z = 5
We manipulate the equations in such a way that both x and y can be expressed in terms of z. To do so, we need two steps:
a) We first try to eliminate x from the second equation. By subtracting the first equation
Figure 1.4: Sloping walls.
x + y + z = 3 −y − 3z = −1
b) Next, we add the new second equation to the first one in order to get rid of the variable y in the first equation. We find
x − 2z = 2 −y − 3z = −1
Now, both x and y can be expressed in terms of z as follows: x = 2 + 2z and y = 1 − 3z. Introduce a parameter λ by z = λ. Then we get:
x = 2 + 2λ y = 1 − 3λ z = λ.
Separating the ‘constant part’ and the ‘variable part’, we usually rewrite this as
(x, y, z) = (2, 1, 0) + λ(2,−3, 1).
6 Shapes in architecture
This notation suggests clearly that, not surprisingly, we are dealing with a line: start at the point (2, 1, 0) and move from there in the direction of (2,−3, 1) by varying λ.
The angle between two planes
From the equations x + y + z = 3 and 2x + y − z = 5, the angle φ between the planes can be computed. The relevant information is contained in the coefficients of x, y and z of both equations (the coefficients 3 and 5 on the right–hand side are irrelevant). The three coefficients of the first equation lead to (1, 1, 1). It turns out that the direction from (0, 0, 0) to (1, 1, 1) is perpendicular to the first plane (more on this in Chapter 2). Likewise, the three coefficients of the second equation lead to (2, 1,−1), and the direction from (0, 0, 0) to (2, 1,−1) is perpendicular to the second plane. In this setting, where directions come into play, we usually speak of vectors. The angle between the two planes equals the angle between these two ‘vectors’ (make a picture to convince yourselves). It turns out (and this will be dealt with more extensively in Chapter 2) that the cosine of the angle is computed as follows from the two vectors (1, 1, 1) and (2, 1,−1):
cos φ = 1 · 2 + 1 · 1 + 1 · (−1)√
12 + 12 + 12 · √
22 + 12 + (−1)2 =
√ 6
= 2
3 √
2 =
√ 2
3 .
From this expression, the angle is easily found to be approximately 1.08 radians or 61.9 degrees.
Varying the plane
If we replace one or more of the coefficients in the equation of one of the planes by an (expression in an) auxiliary parameter, we obtain a varying family of planes : for each value of the parameter a plane is defined. Such a family may be useful in the design of a building where you have to specify the position of a wall, say, given that it passes through certain points, intersects the ground level along a certain line, etc. Here are a few examples.
For every value of a the equation x + y + z = a describes a plane. All these planes are parallel to one another (they are all perpendicular to the vector (1, 1, 1)). The plane with equation x+y+z = 0 (i.e., a = 0) contains the origin (0, 0, 0), but the plane with equation x + y + z = 1 evidently does not. You might look for a plane in this family which touches a sphere with center in the origin and given radius.
Here is a family with other properties. The family of planes x + y + az = 3 all pass through the point (3, 0, 0), but no two of them are parallel. They all have the line of intersection of the two planes z = 0 and x + y = 3 in common. Among these planes you might be looking for one which makes an angle of 60 with a horizontal plane.
Rotations and translations are also important ways of varying a plane; these operations will be discussed in later chapters.
1.1.5 Buildings with curved exteriors
Modern buildings show a variety of curved shapes, like the Gherkin in London, see Fig. (1.5)2. To handle these, nonlinear equations and nonlinear parametric descriptions…
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