Reflection, Mirrors and Lenses | Physics

Learning objectives

Teacher Support

The learning objectives in this section help your students master the following standards:

(7) Science concepts. The student knows the characteristics and behavior of waves. The student is expected to:
- (D) investigate behaviors of waves, including reflection, refraction, diffraction, interference, resonance, and the Doppler effect;
- (E) describe and predict image formation as a consequence of reflection from a plane mirror and refraction through a thin convex lens; and
- (F) describe the role of wave characteristics and behaviors in medical and industrial applications.

In addition, the High School Physics Laboratory Manual addresses content in this section in the lab titled: Mirrors and Lenses, as well as the following standards:

(7) Science concepts. The student knows the characteristics and behavior of waves. The student is expected to:
- (D) investigate behaviors of waves, including reflection, refraction, diffraction, interference, resonance, and the Doppler effect.

Section Key Terms

angle of incidence	angle of reflection	central axis	concave mirror	convex mirror
diffused	focal length	focal point	geometric optics	law of reflection
law of refraction	ray	real image	specular	virtual image

Characteristics of Mirrors

There are three ways, as shown in Figure_16_01_Ray, in which light can travel from a source to another location. It can come directly from the source through empty space, such as from the Sun to Earth. Light can travel to an object through various media, such as air and glass. Light can also arrive at an object after being reflected, such as by a mirror. In all these cases, light is modeled as traveling in a straight line, called a ray. Light may change direction when it encounters the surface of a different material (such as a mirror) or when it passes from one material to another (such as when passing from air into glass). It then continues in a straight line—that is, as a ray. The word ray comes from mathematics. Here it means a straight line that originates from some point. It is acceptable to visualize light rays as laser rays (or even science fiction depictions of ray guns).

View original figure (external; image not reproduced in this text edition)

Original figure description: In view (a), a cartoon image of Earth contains a house superimposed over North America. The Sun is included above and to the right of Earth. An arrow points from the Sun’s rays to a series of lines, which represent Earth’s atmosphere, directly above the Earth. In view (b), a woman is looking out of a window at her car. Arrows show the Sun’s rays bouncing off the car and traveling as light toward the woman.

Three methods for light to travel from a source to another location are shown. (a) Light reaches the upper atmosphere of Earth by traveling through empty space directly from the source (the Sun). (b) This light can reach a person in one of two ways. It can travel through a medium, such as air or glass, and typically travels from one medium to another. It can also reflect from an object, such as a mirror.

Because light moves in straight lines, that is, as rays, and changes directions when it interacts with matter, it can be described through geometry and trigonometry. This part of optics, described by straight lines and angles, is therefore called geometric optics. There are two laws that govern how light changes direction when it interacts with matter: the law of reflection, for situations in which light bounces off matter; and the law of refraction, for situations in which light passes through matter. In this section, we consider the geometric optics of reflection.

Teacher Support

[BL]Explain that light bounces is a simplification. The geometry of the path of a bouncing ball is similar to that of light, but what happens at the point of impact is different at the molecular level.

[OL]Indicate that the terms right angle, perpendicular, and normal line all mean the same thing: a vertical line at a 90° angle to a flat surface.

[AL]Recount and explain all the possible interactions of light with matter. Light can be absorbed at the surface of an opaque object. Some colors of light may be absorbed and others reflected. Light often is partially absorbed and partially reflected. It may also be transmitted through a transparent material, such as water or glass. Typically, if the surface of a transparent material is smooth, such as that of a window pane, light is transmitted partially and reflected partially.

Whenever we look into a mirror or squint at sunlight glinting from a lake, we are seeing a reflection. How does the reflected light travel from the object to your eyes? The law of reflection states: The angle of reflection, $θ_{r}$ , equals the angle of incidence, $θ_{i}$ . This law governs the behavior of all waves when they interact with a smooth surface, and therefore describe the behavior of light waves as well. The reflection of light is simplified when light is treated as a ray. This concept is illustrated in Figure_16_01_Reflect, which also shows how the angles are measured relative to the line perpendicular to the surface at the point where the light ray strikes it. This perpendicular line is also called the normal line, or just the normal. Light reflected in this way is referred to as specular (from the Latin word for mirror: speculum).

We expect to see reflections from smooth surfaces, but Figure_16_01_Reflect2, illustrates how a rough surface reflects light. Because the light is reflected from different parts of the surface at different angles, the rays go in many different directions, so the reflected light is diffused. Diffused light allows you to read a printed page from almost any angle because some of the rays go in different directions. Many objects, such as people, clothing, leaves, and walls, have rough surfaces and can be seen from many angles. A mirror, on the other hand, has a smooth surface and reflects light at specific angles.

View original figure (external; image not reproduced in this text edition)

Original figure description: A smooth, flat reflective surface is shown. A dotted line drops from the top of the figure and ends when it reaches the smooth surface. This line is perpendicular to the surface; there is a 90° angle between the surface and the perpendicular line. To the left of the perpendicular line, a ray (called an “incident ray” strikes the surface at a 45° angle. This ray—the “reflective ray”—is reflected off the surface to the right of the perpendicular line, again at a 45° angle. The angle between the incident ray and the perpendicular line is θi; the angle between the perpendicular line and the reflected ray is θr.

The law of reflection states that the angle of reflection, θ_r, equals the angle of incidence, θ_i. The angles are measured relative to the line perpendicular to the surface at the point where the ray strikes the surface. The incident and reflected rays, along with the normal, lie in the same plane.

View original figure (external; image not reproduced in this text edition)

Original figure description: A rough surface, such as the ground, is shown. Multiple rays are striking the uneven surface and are bouncing off the ground in different directions.

Light is diffused when it reflects from a rough surface. Here, many parallel rays are incident, but they are reflected at many different angles because the surface is rough.

When we see ourselves in a mirror, it appears that our image is actually behind the mirror. We see the light coming from a direction determined by the law of reflection. The angles are such that our image is exactly the same distance behind the mirror, d_i, as the distance we stand away from the mirror, d_o. Although these mirror images make objects appear to be where they cannot be (such as behind a solid wall), the images are not figments of our imagination. Mirror images can be photographed and videotaped by instruments and look just as they do to our eyes, which are themselves optical instruments. An image in a mirror is said to be a virtual image, as opposed to a real image. A virtual image is formed when light rays appear to diverge from a point without actually doing so.

Figure_16_01_Flat_mirror helps illustrate how a flat mirror forms an image. Two rays are shown emerging from the same point, striking the mirror, and reflecting into the observer’s eye. The rays can diverge slightly, and both still enter the eye. If the rays are extrapolated backward, they seem to originate from a common point behind the mirror, allowing us to locate the image. The paths of the reflected rays into the eye are the same as if they had come directly from that point behind the mirror. Using the law of reflection—the angle of reflection equals the angle of incidence—we can see that the image and object are the same distance from the mirror. This is a virtual image, as defined earlier.

View original figure (external; image not reproduced in this text edition)

Original figure description: A flat mirror is placed perpendicular to our field of view. We see the side of the mirror. A bottle is placed to the left of the mirror at distance do to the mirror. Two rays originate from the bottle, strike the mirror, and reflect to an eye that is looking at the mirror. To the right of the mirror, a virtual image of the bottle is included as a backward extrapolation of the path of the rays from the eye to the mirror. The bottle to the right of the mirror is at distance di from the mirror.

When two sets of rays from common points on an object are reflected by a flat mirror into the eye of an observer, the reflected rays seem to originate from behind the mirror, which determines the position of the virtual image.

Mirror Mazes

Figure_16_01_Maze is a chase scene from an old silent film called The Circus, starring Charlie Chaplin. The chase scene takes place in a mirror maze. You may have seen such a maze at an amusement park or carnival. Finding your way through the maze can be very difficult. Keep in mind that only one image in the picture is real—the others are virtual.

View original figure (external; image not reproduced in this text edition)

Original figure description: This black-and-white still shot shows Charlie Chaplin and a thief in a maze of mirrors. Images of Chaplin and the thief appear in mirrors, within mirrors, within mirrors.

Charlie Chaplin is in a mirror maze. Which image is real?

One of the earliest uses of mirrors for creating the illusion of space is seen in the Palace of Versailles, the former home of French royalty. Construction of the Hall of Mirrors (Figure_16_01_Versailles) began in 1678. It is still one of the most popular tourist attractions at Versailles.

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Original figure description: The Hall of Mirrors at Versailles features an al fresco and gilded domed ceiling; many large, hanging crystal chandeliers; exquisite painted panels on the walls; and mirrors—lots and lots of mirrors—to give an illusion of space. Tourists are pictured roaming the Hall.

Tourists love to wander in the Hall of Mirrors at the Palace of Versailles. (credit: Michal Osmenda, Flickr)

Only one Charlie in this image (fs-id1164566710020) is real. The others are all virtual images of him. Can you tell which is real? Hint—His hat is tilted to one side.

View original figure (external; image not reproduced in this text edition)

Original figure description: A grayscale image of Charlie Chaplain in a house of mirrors is shown. There are infinite images of him, His image in the foreground has his hat tilted to the left.

The virtual images have their hats tilted to the right.
The virtual images have their hats tilted to the left.
The real images have their hats tilted to the right.
The real images have their hats tilted to the left.

Virtual Image

This video explains the creation of virtual images in a mirror. It shows the location and orientation of the images using ray diagrams, and relates the perception to the human eye.

Virtual image | Geometric optics | Physics | Khan Academy

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Some mirrors are curved instead of flat. A mirror that curves inward is called a concave mirror, whereas one that curves outward is called a convex mirror. Pick up a well-polished metal spoon and you can see an example of each type of curvature. The side of the spoon that holds the food is a concave mirror; the back of the spoon is a convex mirror. Observe your image on both sides of the spoon.

Ray diagrams can be used to find the point where reflected rays converge or appear to converge, or the point from which rays appear to diverge. This is called the focal point, F. The distance from F to the mirror along the central axis (the line perpendicular to the center of the mirror’s surface) is called the focal length, f. Figure_16_01_Curved_mirror shows the focal points of concave and convex mirrors.

View original figure (external; image not reproduced in this text edition)

Original figure description: There are two curved mirrors in this figure. In view (a), the front of a concave mirror is struck by rays of light that originate in front of the mirror. The rays reflect off the concave mirror and converge at varying angles to a point F in front of the mirror. The distance from F to the mirror is f. In view (b), the back of a convex mirror is struck by rays of light that diverge off the back of the mirror at varying angles. If one were to extrapolate the diverging rays backward, to the front of the mirror, the backward-extrapolated rays meet at point F; however, the distance from the back of the mirror to F is a negative f.

(a, b) The focal length for the concave mirror in (a), formed by converging rays, is in front of the mirror, and has a positive value. The focal length for the convex mirror in (b), formed by diverging rays, appears to be behind the mirror, and has a negative value.

Images formed by a concave mirror vary, depending on which side of the focal point the object is placed. For any object placed on the far side of the focal point with respect to the mirror, the rays converge in front of the mirror to form a real image, which can be projected onto a surface, such as a screen or sheet of paper However, for an object located inside the focal point with respect to the concave mirror, the image is virtual. For a convex mirror the image is always virtual—that is, it appears to be behind the mirror. The ray diagrams in Figure_16_01_Curved2 show how to determine the nature of the image formed by concave and convex mirrors.

View original figure (external; image not reproduced in this text edition)

Original figure description: View (a) shows a concave mirror to the right with a lit candle to the left placed outside the focal point. It is a real image. View (b) shows a concave mirror to the right with a lit candle to the left placed inside the focal point. This image is virtual. View (c) shows a convex mirror to the right with a lit candle to the left. The image is virtual.

(a) The image of an object placed outside the focal point of a concave mirror is inverted and real. (b) The image of an object placed inside the focal point of a concave mirror is erect and virtual. (c) The image of an object formed by a convex mirror is erect and virtual.

The information in Figure_16_01_Curved2 is summarized in Table_16_01.

This table details the type and orientation of images formed by concave and convex mirrors.

Curved Mirror Images
Type of Mirror	Object to Mirror Distance, d_o	Image Characteristics
Concave	$d_{o} > f$	Real and inverted
Concave	$d_{o} < f$	Virtual and erect
Convex	$d_{o} < or > f$	Virtual and erect

Concave and Convex Mirrors

Silver spoon and silver polish, or a new spoon made of any shiny metal

Instructions

Procedure

Choose any small object with a top and a bottom, such as a short nail or tack, or a coin, such as a quarter. Observe the object’s reflection on the back of the spoon.
Observe the reflection of the object on the front (bowl side) of the spoon when held away from the spoon at a distance of several inches.
Observe the image while slowly moving the small object toward the bowl of the spoon. Continue until the object is all the way inside the bowl of the spoon.
You should see one point where the object disappears and then reappears. This is the focal point.

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You should be able to notice everyday applications of curved mirrors. One common example is the use of security mirrors in stores, as shown in Figure_16_01_Convex_Example.

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Original figure description: The image shows a woman taking a picture in front of a convex mirror.

Security mirrors are convex, producing a smaller, upright image. Because the image is smaller, a larger area is imaged compared with what would be observed for a flat mirror; hence, security is improved. (credit: Laura D’Alessandro, Flickr)

Some telescopes also use curved mirrors and no lenses (except in the eyepieces) both to magnify images and to change the path of light. Figure_16_01_Schmidt shows a Schmidt-Cassegrain telescope. This design uses a spherical primary concave mirror and a convex secondary mirror. The image is projected onto the focal plane by light passing through the perforated primary mirror. The effective focal length of such a telescope is the focal length of the primary mirror multiplied by the magnification of the secondary mirror. The result is a telescope with a focal length much greater than the length of the telescope itself.

View original figure (external; image not reproduced in this text edition)

Original figure description: There is an eye to the right of the diagram. The eye represents the focal plane. Moving from right to left, in front of the eye, there is a perforated primary mirror and a secondary mirror. The end of the telescope is to the left of the figure. Rays enter the end of the telescoped, hit the primary mirror, bounce backward to the secondary mirror, bounce through the hole (perforation) in the primary mirror, and travel to the eye.

This diagram shows the design of a Schmidt–Cassegrain telescope.

A parabolic concave mirror has the very useful property that all light from a distant source, on reflection by the mirror surface, is directed to the focal point. Likewise, a light source placed at the focal point directs all the light it emits in parallel lines away from the mirror. This case is illustrated by the ray diagram in Figure_16_01_Concave. The light source in a car headlight, for example, is located at the focal point of a parabolic mirror.

View original figure (external; image not reproduced in this text edition)

Original figure description: This diagram shows a car headlight, also called a spotlight, which has a light bulb surrounded by a parabolic mirror. The mirror is bowl shaped, with the bulb sitting at the center of recessed side of the bowl. Arrows show that light rays from the bulb extend out in all directions and then bounce off of the mirror so that all of the light rays travel straight out from the headlight

The bulb in this ray diagram of a car headlight is located at the focal point of a parabolic mirror.

Parabolic mirrors are also used to collect sunlight and direct it to a focal point, where it is transformed into heat, which in turn can be used to generate electricity. This application is shown in Figure_16_01_Concave2.

View original figure (external; image not reproduced in this text edition)

Original figure description: This photo depicts a long length of parabolic trough deflectors angled toward the sun.

Parabolic trough collectors are used to generate electricity in southern California. (credit: kjkolb, Wikimedia Commons)

The Application of the Curved Mirror Equations

Curved mirrors and the images they create involve a fairly small number of variables: the mirror’s radius of curvature, R; the focal length, f; the distances of the object and image from the mirror, d_o and d_i, respectively; and the heights of the object and image, h_o and h_i, respectively. The signs of these values indicate whether the image is inverted, erect (upright), real, or virtual. We now look at the equations that relate these variables and apply them to everyday problems.

Figure_16_01_Ray2 shows the meanings of most of the variables we will use for calculations involving curved mirrors.

View original figure (external; image not reproduced in this text edition)

Original figure description: This figure shows an old-fashioned key inside the focal point of a concave mirror. All the variables used for calculations involving curved mirrors depicted.

Look for the variables, d_o, d_i, h_o, h_i, and f in this figure.

The basic equation that describes both lenses and mirrors is the lens/mirror equation

\frac{1}{f} = \frac{1}{d_{i}} + \frac{1}{d_{o}} .

This equation can be rearranged several ways. For example, it may be written to solve for focal length.

f = \frac{d_{i} d_{o}}{d_{o} + d_{i}}

Magnification, m, is the ratio of the size of the image, h_i, to the size of the object, h_o. The value of m can be calculated in two ways.

m = \frac{h_{i}}{h_{o}} = \frac{- d_{i}}{d_{o}}

This relationship can be written to solve for any of the variables involved. For example, the height of the image is given by

h_{i} = - h_{o} (\frac{d_{i}}{d_{o}}) .

We saved the simplest equation for last. The radius of curvature of a curved mirror, R, is simply twice the focal length.

R = 2 f

We can learn important information from the algebraic sign of the result of a calculation using the previous equations:

A negative d_i indicates a virtual image; a positive value indicates a real image
A negative h_i indicates an inverted image; a positive value indicates an erect image
For concave mirrors, f is positive; for convex mirrors, f is negative

Now let’s apply these equations to solve some problems.

Calculating Focal Length

A person standing 6.0 m from a convex security mirror forms a virtual image that appears to be 1.0 m behind the mirror. What is the focal length of the mirror?

Solution

$f = \frac{d_{i} d_{o}}{d_{o} + d_{i}} = \frac{(- 1.0) (6.0)}{6.0 + (- 1.0)} = \frac{- 6.0}{5.0} = - 1.2 m$

Discussion

The negative result is expected for a convex mirror. This indicates the focal point is behind the mirror.

Calculating Object Distance

Electric room heaters use a concave mirror to reflect infrared (IR) radiation from hot coils. Note that IR radiation follows the same law of reflection as visible light. Given that the mirror has a radius of curvature of 50.0 cm and produces an image of the coils 3.00 m in front of the mirror, where are the coils with respect to the mirror?

Solution

Because d_i and f are known, the lens/mirror equation can be used to find d_o.

\frac{1}{f} = \frac{1}{d_{i}} + \frac{1}{d_{o}}

Rearranging to solve for d_o, we have

d_{o} = \frac{d_{i} f}{d_{i} - f} .

Entering the known quantities gives us

d_{o} = \frac{(3.00 \times 10^{2}) (25.0)}{(3.00 \times 10^{2}) - 25.0} = 27.3 cm .

Discussion

Note that the object (the coil filament) is farther from the mirror than the mirror’s focal length. This is a case 1 image (d_o > f and f positive), consistent with the fact that a real image is formed. You get the most concentrated thermal energy directly in front of the mirror and 3.00 m away from it. In general, this is not desirable because it could cause burns. Usually, you want the rays to emerge parallel, and this is accomplished by having the filament at the focal point of the mirror.

Note that the filament here is not much farther from the mirror than the focal length, and that the image produced is considerably farther away.

Practice Problems

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What is the focal length of a makeup mirror that produces a magnification of 1.50 when a person’s face is 12.0 cm away? Construct a ray diagram using paper, a pencil and a ruler to confirm your calculation.

–36.0 cm
–7.20 cm
7.20 cm
36.0 cm

Check Your Understanding

How does the object distance, d_o, compare with the focal length, f, for a concave mirror that produces an image that is real and inverted?

d_o > f, where d_o and f are object distance and focal length, respectively.
d_o < f, where d_o and f are object distance and focal length, respectively.
d_o = f, where do and f are object distance and focal length, respectively.
d_o = 0, where do is the object distance.

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Section Summary

The angle of reflection equals the angle of incidence.
Plane mirrors and convex mirrors reflect virtual, erect images. Concave mirrors reflect light to form real, inverted images or virtual, erect images, depending on the location of the object.
Image distance, height, and other characteristics can be calculated using the lens/mirror equation and the magnification equation.

Key Equations

lens/mirror equation (reciprocal version)	$\frac{1}{f} = \frac{1}{d_{i}} + \frac{1}{d_{o}}$
lens/mirror equation (solved version)	$f = \frac{d_{i} d_{o}}{d_{o} + d_{i}}$
magnification equation	$m = \frac{h_{i}}{h_{o}} = \frac{- d_{i}}{d_{o}}$
radius/focal length equation	R = 2f

Concept Items

Part A. Can you see a virtual image? Part B. Can you photograph one? Explain your answers.

A. yes; B. No, an image from a flat mirror cannot be photographed.
A. no; B. Yes, an image from a flat mirror can be photographed.
A. yes; B. Yes, an image from a flat mirror can be photographed.
A. no; B. No, an image from a flat mirror cannot be photographed.

State the law of reflection.

$ϑ_{r} = ϑ_{i}$ , where $ϑ_{r}$ is the angle of reflection and $ϑ_{i}$ is the angle of incidence.
$ϑ_{r} > ϑ_{i}$ , where $ϑ_{r}$ is the angle of reflection and $ϑ_{i}$ is the angle of incidence.
$ϑ_{r} < ϑ_{i}$ , where $ϑ_{r}$ is the angle of reflection and $ϑ_{i}$ is the angle of incidence.
$ϑ_{r} = 0$ , where $ϑ_{r}$ is the angle of reflection.

Critical Thinking

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Problems

Some telephoto cameras use a mirror rather than a lens. What radius of curvature is needed for a concave mirror to replace a 0.800 -m focal-length telephoto lens?

0.400 m
1.60 m
4.00 m
16.0 m

What is the focal length of a makeup mirror that produces a magnification of 2.00 when a person’s face is 8.00 cm away?

–16 cm
–5.3 cm
5.3 cm
16 cm

Test Prep Multiple Choice

In geometric optics, a straight line emerging from a point is called a (an) ________.

ray
focal point
image
object distance

An image of a 2.0 -cm object reflected from a mirror is 5.0 cm tall. What is the magnification of the mirror?

Can a virtual image be projected onto a screen with additional lenses or mirrors? Explain your answer.

Yes, the rays actually meet behind the lens or mirror.
No, the image is formed by rays that converge to a point in front of the mirror or lens.
Yes, any image that can be seen can be manipulated so that it can be projected onto a screen.
No, the image can only be perceived as being behind the lens or mirror.

Test Prep Short Answer

Distinguish between reflection and refraction in terms of how a light ray changes when it meets the interface between two media.

Reflected light penetrates the surface whereas refracted light is bent as it travels from one medium to the other.
Reflected light penetrates the surface whereas refracted light travels along a curved path.
Reflected light bounces from the surface whereas refracted light travels along a curved path.
Reflected light bounces from the surface whereas refracted light is bent as it travels from one medium to the other.

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A concave mirror has a focal length of 5.00 cm. What is the image distance of an object placed 7.00 cm from the center of the mirror?

−17.5 cm
−2.92 cm
2.92 cm
17.5 cm

An 8.0 -cm tall object is placed 6.0 cm from a concave mirror with a magnification of 2.0. What are the image height and the image distance?

h_i = 16 cm, d_i = 12 cm
h_i = 16 cm, d_i = 12 cm
h_i = 16 cm, d_i = 12 cm
h_i = 16 cm, d_i = 12 cm

Test Prep Extended Response

The diagram shows a lightbulb between two mirrors. One mirror produces a beam of light with parallel rays; the other keeps light from escaping without being put into the beam.

View original figure (external; image not reproduced in this text edition)

Original figure description: A diagram illustrating a light bulb positioned between two concave mirrors, demonstrating how light rays are reflected to create a parallel beam, often used in headlights or flashlights.

Where is the light source in relation to the focal point or radius of curvature of each mirror? Explain your answer.

The bulb is at the center of curvature of the small mirror and at the focal point of the large mirror.
The bulb is at the focal point of the small mirror and at the focal point of the large mirror.
The bulb is at the center of curvature of the small mirror and at the center of curvature of the large mirror.
The bulb is at the focal point of the small mirror and at the center of curvature of the large mirror.

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Glossary

angle of incidence: the angle, with respect to the normal, at which a ray meets a boundary between media or a reflective surface
angle of reflection: the angle, with respect to the normal, at which a ray leaves a reflective surface
central axis: a line perpendicular to the center of a lens or mirror extending in both directions
concave mirror: a mirror with a reflective side that is curved inward
convex mirror: a mirror with a reflective side that is curved outward
focal length: the distance from the focal point to the mirror
focal point: the point at which rays converge or appear to converge
law of reflection: the law that indicates the angle of reflection equals the angle of incidence
law of refraction: the law that describes the relationship between refractive indices of materials on both sides of a boundary and the change in the path of light crossing the boundary, as given by the equation n₁ sin $θ_{1}$ = n₂ sin $θ_{2}$
ray: light traveling in a straight line
real image: an optical image formed when light rays converge and pass through the image, producing an image that can be projected onto a screen
virtual image: the point from which light rays appear to diverge without actually doing so

Attribution. Physics, OpenStax, Rice University, and the contributing authors credited in the book. Licensed under Creative Commons Attribution 4.0 International. Source revision 56e71824e7 of openstax/osbooks-physics (2026-10-13). CC BY 4.0. Original source ↗. Text edition of a pinned CC BY 4.0 source revision of the OpenStax book, converted from CNXML by the EvidenceHelix importer. Original prose, learning objectives, tables, exercises, glossary and captions are preserved; images are not copied (figures link to the original source and keep their credits). No new content was added. Later revisions of this book can carry different licence terms and are not used.