What is great about Euler's formula is that it can be understood by almost anyone, as it is so simple to write
down. Euler's formula can be understood by someone in Year 7, but is also interesting enough to be studied
in universities as part of the mathematical area called topology.
Euler's formula deals with shapes called Polyhedra. A Polyhedron is a closed solid shape which has flat faces and straight edges. An example of a polyhedron would be a cube, whereas a cylinder is not a polyhedron as it has curved edges.
Euler's formula states that for Polyhedra that follow certain rules:
F+V-E=2
Where F= Number of faces, V= Number of vertices (Corners), E=Number of edges
Let us check that this works for the Tetrahedron (triangle based pyramid) and the cube.
Shape Faces Vertices Edges F+V-E
Tetrahedron 4 4 6 4+4-6=2
Cube 6 8 12 6+8-12=2
Euler's Formula does however only work for Polyhedra that follow certain rules. The rule is that the shape
must not have any holes, and that it must not intersect itself. (Imagine taking two opposite faces on a shape
and gluing them together at a particular point. This is not allowed.) It also cannot be made up of two pieces
stuck together, such as two cubes stuck together by one vertex. If none of these rules are broken, then
F+V-E=2 for all Polyhedra. Euler’s formula works for most of the common polyhedra which we have heard
of..
There are in fact shapes which produce a different answer to the sum F+V-E. The answer to the sum F+VE is called the Euler Characteristic χ, and is often written F+V-E=χ . Some shapes can even have an Euler Characteristic which is negative! This is where it can all start to get quite complicated. If you want to find about the Euler Characteristic of some of the strange shapes below, why not take a look here.
Who was Euler?
Leonhard Euler (1707-1783) was a Swiss mathematician who is thought to be one of the greatest and most
productive mathematicians of all time. Euler spent much of his career blind, but losing his sight only seemed
to make him even more productive and at one point he was writing one paper per week, with scribes writing
his work down for him. There is more than one formula named after Euler, and the one we have just looked
at is sometimes called Euler’s Polyhedral formula. Euler’s other formula is in the field of complex
numbers. Euler is pronounced ‘Oiler’.
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VECTORS
Vectors are a relatively new arrival to the world of mathematics, dating only from the 19th century. They provide us with some elegant and powerful techniques for computing angles between lines and the orientation of surfaces. They also provide a coherent framework for computing the behaviour of dynamic objects in computer animation and illumination models in rendering. We often employ a single number to represent quantities that we use in our daily lives such as, height, age, shoe size, waist and chest measurements. The magnitude of this number depends on our age and whether we use metric or imperial units. Such quantities are called scalars. In computer graphics scalar quantities include colour, height, width, depth, brightness, number of frames, etc. On the other hand, there are some things that require more than one number to represent them: wind, force, weight, velocity and sound are just a few examples. These cannot be represented accurately by a single number. For example, any sailor knows that wind has a magnitude and a direction. The force we use to lift an object also has a value and a direction. Similarly, the velocity of a moving object is measured in terms of its speed (e.g. miles per hour) and a direction such as north-west. Sound, too, has intensity and a direction. These quantities are called vectors. In computer graphics, vectors are generally made of two or three numbers, and this is the only type we will consider in this chapter. Mathematicians such as Caspar Wessel (1745–1818), Jean Argand (1768– 1822) and John Warren (1796–1852) were simultaneously exploring complex numbers and their graphical representation. In 1837, Sir William Rowan Hamilton (1788–1856) made his breakthrough with quaternions. In 1853, Hamilton published his book Lectures on Quaternions in which he described terms such as vector, transvector and provector. Hamilton’s work was not widely accepted until 1881, when the American mathematician Josiah Gibbs (1839–1903) published his treatise Vector Analysis, describing modern vector analysis. 6.1 2D Vectors In computer graphics we employ 2D and 3D vectors. In this chapter we first consider vector notation in a 2D context and then extrapolate the ideas into 3D. 6.1.1 Vector Notation A scalar such as x is just a name for a single numeric quantity. However, because a vector contains two or more numbers, its symbolic name is printed using a bold font to distinguish it from a scalar variable. Examples are n, i and Q. When a scalar variable is assigned a value we employ the standard algebraic notation x = 3 However, when a vector is assigned its numeric values,
the following notation is used: which is called a column vector. The numbers 3 and 4 are called the components of n, and their position within the brackets is significant. A row vector transposes the components horizontally, n = [3 4]T where the superscriptT reminds us of the transposition. 6.1.2 Graphical Representation of Vectors Because vectors have to encode direction as well as magnitude, an arrow could be used to indicate direction and a number to specify magnitude. Such a scheme is often used in weather maps. Although this is a useful graphical interpretation for such data, it is not practical for algebraic manipulation. Cartesian coordinates provide an excellent mechanism for visualizing vectors and allowing them to be incorporated within the classical framework of mathematics. Figure 6.1 shows a vector represented by a short line segment. The length of the line represents the vector’s magnitude, and the orientation defines its direction. But as you can see from the figure, the line does not have a direction. Even if we attach an arrowhead to the line, which is standard practice for annotating vectors in books and scientific papers, the arrowhead has no mathematical reality.
Graphical Representation of Vectors :Because vectors have to encode direction as well as magnitude, an arrow could be used to indicate direction and a number to specify magnitude. Such a scheme is often used in weather maps. Although this is a useful graphical interpretation for such data, it is not practical for algebraic manipulation. Cartesian coordinates provide an excellent mechanism for visualizing vectors and allowing them to be incorporated within the classical framework of mathematics. Figure 6.1 shows a vector represented by a short line segment. The length of the line represents the vector’s magnitude, and the orientation defines its direction. But as you can see from the figure, the line does not have a direction. Even if we attach an arrowhead to the line, which is standard practice for annotating vectors in books and scientific papers, the arrowhead has no mathematical reality.
Fig. 6.1. A vector represented by a short line segment. However, although the vector has magnitude, it does not have direction.
Fig. 6.2. Two vectors r and s have the same magnitude and opposite directions.
The line’s direction can be determined by first identifying the vector’s tail and then measuring its components along the x - and y-axes. For example, in Figure 6.2 the vector r has its tail defined by (x1, y1) = (1, 2) and its head by (x2, y2) = (2, 3). Vector s, on the other hand, has its tail defined by (x3, y3) = (2, 2) and its head by (x4, y4) = (1, 1). The x - and y-components for r are computed as follows:
xr = (x2 − x1) yr = (y2 − y1) xr = 2 − 1=1 yr = 3 − 2=1 whereas the components for s are computed as follows: xs = (x4 − x3) ys = (y4 − y3) xs = 1 − 2 = −1 ys = 1 − 2 = −1 xs = −1 ys = −1 It is the negative values of xs and ys that encode the vector’s direction. In general, given that the coordinates of a vector’s head and tail are (xh, yh) and
Fig. 6.3. Eight vectors, whose coordinates are shown in Table 6.1.
(xt, yt) respectively, its components Δx and Δy are given by Δx = (xh − xt) Δy = (yh − yt) (6.1) One can readily see from this notation that a vector does not have a unique position in space. It does not matter where we place a vector: so long as we preserve its length and orientation, its components will not alter.
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Magnitude of a Vector:
The magnitude of a vector r is expressed by and is computed by applying the theorem of
Pythagoras to its components:
To illustrate these ideas, consider a vector defined by (xh, yh) = (3, 4) and (xt, yt) = (1, 1). The x - and y-components are 2 and 3 respectively. Therefore its magnitude is equal to Figure 6.3 shows various vectors, and their properties are listed in Table 6.1.
6.2 3D Vectors The above vector examples are in 2D, but it is extremely simple to extend this notation to embrace an extra dimension. Figure 6.4 shows a 3D vector r with its head, tail, components and magnitude annotated. The components and magnitude are given by Δx = (xh − xt) (6.3)
Fig. 6.4. The 3D vector has components Δx, Δy, Δz, which are the differences between the head and tail coordinates.
down. Euler's formula can be understood by someone in Year 7, but is also interesting enough to be studied
in universities as part of the mathematical area called topology.
Euler's formula deals with shapes called Polyhedra. A Polyhedron is a closed solid shape which has flat faces and straight edges. An example of a polyhedron would be a cube, whereas a cylinder is not a polyhedron as it has curved edges.
Euler's formula states that for Polyhedra that follow certain rules:F+V-E=2
Where F= Number of faces, V= Number of vertices (Corners), E=Number of edges
Let us check that this works for the Tetrahedron (triangle based pyramid) and the cube.
Shape Faces Vertices Edges F+V-ETetrahedron 4 4 6 4+4-6=2
Cube 6 8 12 6+8-12=2
Euler's Formula does however only work for Polyhedra that follow certain rules. The rule is that the shape
must not have any holes, and that it must not intersect itself. (Imagine taking two opposite faces on a shape
and gluing them together at a particular point. This is not allowed.) It also cannot be made up of two pieces
stuck together, such as two cubes stuck together by one vertex. If none of these rules are broken, then
F+V-E=2 for all Polyhedra. Euler’s formula works for most of the common polyhedra which we have heard
of..
There are in fact shapes which produce a different answer to the sum F+V-E. The answer to the sum F+VE is called the Euler Characteristic χ, and is often written F+V-E=χ . Some shapes can even have an Euler Characteristic which is negative! This is where it can all start to get quite complicated. If you want to find about the Euler Characteristic of some of the strange shapes below, why not take a look here.
Who was Euler?
Leonhard Euler (1707-1783) was a Swiss mathematician who is thought to be one of the greatest and most
productive mathematicians of all time. Euler spent much of his career blind, but losing his sight only seemed
to make him even more productive and at one point he was writing one paper per week, with scribes writing
his work down for him. There is more than one formula named after Euler, and the one we have just looked
at is sometimes called Euler’s Polyhedral formula. Euler’s other formula is in the field of complex
numbers. Euler is pronounced ‘Oiler’.
----------------------------------------------------------------------------------------------------------------------------------------------------------------
VECTORS
Vectors are a relatively new arrival to the world of mathematics, dating only from the 19th century. They provide us with some elegant and powerful techniques for computing angles between lines and the orientation of surfaces. They also provide a coherent framework for computing the behaviour of dynamic objects in computer animation and illumination models in rendering. We often employ a single number to represent quantities that we use in our daily lives such as, height, age, shoe size, waist and chest measurements. The magnitude of this number depends on our age and whether we use metric or imperial units. Such quantities are called scalars. In computer graphics scalar quantities include colour, height, width, depth, brightness, number of frames, etc. On the other hand, there are some things that require more than one number to represent them: wind, force, weight, velocity and sound are just a few examples. These cannot be represented accurately by a single number. For example, any sailor knows that wind has a magnitude and a direction. The force we use to lift an object also has a value and a direction. Similarly, the velocity of a moving object is measured in terms of its speed (e.g. miles per hour) and a direction such as north-west. Sound, too, has intensity and a direction. These quantities are called vectors. In computer graphics, vectors are generally made of two or three numbers, and this is the only type we will consider in this chapter. Mathematicians such as Caspar Wessel (1745–1818), Jean Argand (1768– 1822) and John Warren (1796–1852) were simultaneously exploring complex numbers and their graphical representation. In 1837, Sir William Rowan Hamilton (1788–1856) made his breakthrough with quaternions. In 1853, Hamilton published his book Lectures on Quaternions in which he described terms such as vector, transvector and provector. Hamilton’s work was not widely accepted until 1881, when the American mathematician Josiah Gibbs (1839–1903) published his treatise Vector Analysis, describing modern vector analysis. 6.1 2D Vectors In computer graphics we employ 2D and 3D vectors. In this chapter we first consider vector notation in a 2D context and then extrapolate the ideas into 3D. 6.1.1 Vector Notation A scalar such as x is just a name for a single numeric quantity. However, because a vector contains two or more numbers, its symbolic name is printed using a bold font to distinguish it from a scalar variable. Examples are n, i and Q. When a scalar variable is assigned a value we employ the standard algebraic notation x = 3 However, when a vector is assigned its numeric values,
the following notation is used: which is called a column vector. The numbers 3 and 4 are called the components of n, and their position within the brackets is significant. A row vector transposes the components horizontally, n = [3 4]T where the superscriptT reminds us of the transposition. 6.1.2 Graphical Representation of Vectors Because vectors have to encode direction as well as magnitude, an arrow could be used to indicate direction and a number to specify magnitude. Such a scheme is often used in weather maps. Although this is a useful graphical interpretation for such data, it is not practical for algebraic manipulation. Cartesian coordinates provide an excellent mechanism for visualizing vectors and allowing them to be incorporated within the classical framework of mathematics. Figure 6.1 shows a vector represented by a short line segment. The length of the line represents the vector’s magnitude, and the orientation defines its direction. But as you can see from the figure, the line does not have a direction. Even if we attach an arrowhead to the line, which is standard practice for annotating vectors in books and scientific papers, the arrowhead has no mathematical reality.Graphical Representation of Vectors :Because vectors have to encode direction as well as magnitude, an arrow could be used to indicate direction and a number to specify magnitude. Such a scheme is often used in weather maps. Although this is a useful graphical interpretation for such data, it is not practical for algebraic manipulation. Cartesian coordinates provide an excellent mechanism for visualizing vectors and allowing them to be incorporated within the classical framework of mathematics. Figure 6.1 shows a vector represented by a short line segment. The length of the line represents the vector’s magnitude, and the orientation defines its direction. But as you can see from the figure, the line does not have a direction. Even if we attach an arrowhead to the line, which is standard practice for annotating vectors in books and scientific papers, the arrowhead has no mathematical reality.
Fig. 6.1. A vector represented by a short line segment. However, although the vector has magnitude, it does not have direction.
Fig. 6.2. Two vectors r and s have the same magnitude and opposite directions.
The line’s direction can be determined by first identifying the vector’s tail and then measuring its components along the x - and y-axes. For example, in Figure 6.2 the vector r has its tail defined by (x1, y1) = (1, 2) and its head by (x2, y2) = (2, 3). Vector s, on the other hand, has its tail defined by (x3, y3) = (2, 2) and its head by (x4, y4) = (1, 1). The x - and y-components for r are computed as follows:
xr = (x2 − x1) yr = (y2 − y1) xr = 2 − 1=1 yr = 3 − 2=1 whereas the components for s are computed as follows: xs = (x4 − x3) ys = (y4 − y3) xs = 1 − 2 = −1 ys = 1 − 2 = −1 xs = −1 ys = −1 It is the negative values of xs and ys that encode the vector’s direction. In general, given that the coordinates of a vector’s head and tail are (xh, yh) and
Fig. 6.3. Eight vectors, whose coordinates are shown in Table 6.1.
(xt, yt) respectively, its components Δx and Δy are given by Δx = (xh − xt) Δy = (yh − yt) (6.1) One can readily see from this notation that a vector does not have a unique position in space. It does not matter where we place a vector: so long as we preserve its length and orientation, its components will not alter.
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Magnitude of a Vector:
The magnitude of a vector r is expressed by and is computed by applying the theorem of
Pythagoras to its components:
To illustrate these ideas, consider a vector defined by (xh, yh) = (3, 4) and (xt, yt) = (1, 1). The x - and y-components are 2 and 3 respectively. Therefore its magnitude is equal to Figure 6.3 shows various vectors, and their properties are listed in Table 6.1.
6.2 3D Vectors The above vector examples are in 2D, but it is extremely simple to extend this notation to embrace an extra dimension. Figure 6.4 shows a 3D vector r with its head, tail, components and magnitude annotated. The components and magnitude are given by Δx = (xh − xt) (6.3)
Fig. 6.4. The 3D vector has components Δx, Δy, Δz, which are the differences between the head and tail coordinates.
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