Ray Diagrams & Sign Convention for Mirrors - Tenth Grade Science
Geometric ray tracing provides an analytical and visual method for determining the position, nature, size, and orientation of images formed by curved spherical mirrors. Rather than tracing the infinite number of light rays that radiate from an illuminated object, geometric optics uses four standard reference rays whose reflected trajectories are dictated by the laws of reflection. Furthermore, to translate visual geometric ray diagrams into quantitative algebraic calculations, physicists apply a standardized Cartesian sign convention. In this chapter, you will investigate the four rules of ray tracing for spherical mirrors, trace the six distinct object-image positions for concave mirrors, analyze image formation by convex mirrors, master the New Cartesian Sign Convention, and evaluate the practical applications of each mirror geometry.
The Four Standard Geometric Rules of Ray Tracing
To determine the location and characteristics of an image formed by a spherical mirror, you need to trace at least two light rays originating from a point on the object and find their point of intersection (real image) or apparent divergence (virtual image).
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| THE FOUR STANDARD RULES OF RAY TRACING |
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| RULE 1: PARALLEL RAY REFLECTS THROUGH FOCUS |
| - In Concave: Ray parallel to axis reflects physically THROUGH Focus (F). |
| - In Convex: Ray parallel to axis reflects so it APPEARS to come from F. |
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| RULE 2: FOCAL RAY REFLECTS PARALLEL |
| - In Concave: Ray passing through Focus (F) reflects PARALLEL to axis. |
| - In Convex: Ray directed toward Focus (F) reflects PARALLEL to axis. |
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| RULE 3: CENTER OF CURVATURE RAY RETRACES PATH |
| - In Concave: Ray passing through C strikes normally (i = 0) and RETRACES path. |
| - In Convex: Ray directed toward C strikes normally and RETRACES path. |
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| RULE 4: POLE OBLIQUE RAY REFLECTS SYMMETRICALLY |
| - Ray striking the Pole (P) at angle i reflects at an EQUAL angle r (Angle i = r)|
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The Four Construction Rules Explained
Rule 1 (The Parallel Ray): A ray of light traveling parallel to the principal axis, after reflection from a concave mirror, passes physically through the principal focus ($F$). In a convex mirror, a ray traveling parallel to the principal axis reflects such that it appears to diverge from the principal focus located behind the mirror.
Rule 2 (The Focal Ray): A ray of light passing through the principal focus ($F$) of a concave mirror (or directed toward the focus of a convex mirror), after reflection, emerges parallel to the principal axis. This is the reverse of Rule 1, in accordance with the principle of reversibility of light paths.
Rule 3 (The Central Ray): A ray of light passing through the center of curvature ($C$) of a concave mirror (or directed toward the center of curvature of a convex mirror), after reflection, retraces its path back along the same line. Because any line passing through the center of curvature is a normal to the spherical mirror surface, the angle of incidence is $\angle i = 0^\circ$; by the law of reflection, $\angle r = 0^\circ$, so the ray reflects back on itself.
Rule 4 (The Polar Ray): A ray of light directed obliquely toward the pole ($P$) of a spherical mirror reflects obliquely such that the angle of reflection equals the angle of incidence ($\angle i = \angle r$), with the principal axis serving as the normal.
Ray Tracing and Image Formation by Concave Mirrors
A concave mirror forms real, inverted images or virtual, erect images depending on the distance ($u$) of the object from the mirror pole.
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| SIX OBJECT POSITIONS FOR CONCAVE MIRRORS |
+---------------------+---------------------+---------------------+-----------------+
| Object Position | Image Position | Relative Image Size | Image Nature |
+---------------------+---------------------+---------------------+-----------------+
| 1. At Infinity | At the Focus (F) | Highly diminished, | Real & Inverted |
| | | point-sized | |
| 2. Beyond C | Between F and C | Diminished | Real & Inverted |
| 3. At C | At C | Same size as object | Real & Inverted |
| 4. Between C and F | Beyond C | Magnified (Enlarged)| Real & Inverted |
| 5. At Focus (F) | At Infinity | Highly magnified | Real & Inverted |
| 6. Between P and F | Behind the mirror | Magnified (Enlarged)| Virtual & Erect |
+---------------------+---------------------+---------------------+-----------------+
Case 1: Object at Infinity ($u = \infty$)
When an object is positioned at an infinite distance (such as the Sun or distant stars), incident rays arriving at the mirror are parallel to one another: Parallel incident rays striking the concave mirror reflect and converge at the focal plane. Image Characteristics: Located at the Principal Focus ($F$); extremely diminished (point-sized); real and inverted. Application: Solar furnaces, parabolic dish concentrators, and primary mirrors of astronomical reflecting telescopes.
Case 2: Object Beyond the Center of Curvature ($u > 2f$)
When an object $AB$ is positioned beyond $C$: Ray 1 from the top of the object travels parallel to the axis and reflects through the focus ($F$). Ray 2 passes through the center of curvature ($C$) and retraces its path back through $C$. The two reflected rays intersect between $C$ and $F$ to form image $A'B'$. Image Characteristics: Located between $C$ and $F$; diminished in size; real and inverted.
Case 3: Object at the Center of Curvature ($u = 2f$)
When the object is placed at the center of curvature ($C$): Ray 1 travels parallel to the axis and reflects through $F$. Ray 2 passes through $F$ and reflects parallel to the axis. The reflected rays intersect at $C$ directly below the object. Image Characteristics: Located at $C$; identical in size to the object (magnification $m = -1$); real and inverted.
Case 4: Object Between the Center of Curvature and Focus ($f < u < 2f$)
When the object is placed between $C$ and $F$: Ray 1 parallel to the axis reflects through $F$. Ray 2 passing through $F$ reflects parallel to the axis. The reflected rays intersect beyond $C$. Image Characteristics: Located beyond $C$; magnified (enlarged); real and inverted. Application: Optical projection systems and cinema projectors.
Case 5: Object at the Principal Focus ($u = f$)
When the object is placed at the principal focus ($F$): Ray 1 parallel to the axis reflects through $F$. Ray 2 striking the pole at angle $i$ reflects at an equal angle $r$. The reflected rays are parallel to one another and do not intersect in finite space; they meet at infinity ($v = \infty$). Image Characteristics: Located at infinity; enlarged; real and inverted. Application: Searchlights, automobile headlights, and torches. Placing a bright filament bulb at the focus of a concave reflector produces an intense parallel beam of light.
Case 6: Object Between the Pole and Focus ($u < f$)
When an object is placed close to the mirror, between the pole ($P$) and the focus ($F$): Ray 1 strikes the pole at angle $i$ and reflects at angle $r$. Ray 2 aligned with $C$ reflects back along its normal path. In front of the mirror, the reflected rays diverge away from one another. Tracing the diverging rays backward as dashed lines behind the mirror reveals that they appear to intersect at a point behind the mirror. Image Characteristics: Located behind the mirror; magnified (enlarged); virtual and erect. Application: Shaving mirrors, makeup mirrors, and dental examination mirrors. Placing the face within the focal length produces an enlarged, upright virtual reflection.
Ray Tracing and Image Formation by Convex Mirrors
Regardless of where an object is placed in front of a convex mirror, it forms virtual, erect, and diminished images.
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| IMAGE FORMATION BY A CONVEX MIRROR |
+---------------------+---------------------+---------------------+-----------------+
| Object Position | Image Position | Relative Image Size | Image Nature |
+---------------------+---------------------+---------------------+-----------------+
| 1. At Infinity | Behind mirror at F | Highly diminished, | Virtual & Erect |
| | | point-sized | |
| 2. Anywhere between | Behind mirror | Diminished | Virtual & Erect |
| Infinity and P | between P and F | | |
+---------------------+---------------------+---------------------+-----------------+
Ray Tracing for a Convex Mirror
Place an object $AB$ at a finite distance in front of a convex mirror: Ray 1 travels parallel to the principal axis and reflects outward, appearing to diverge from the principal focus ($F$) behind the mirror. Ray 2 is directed toward the center of curvature ($C$) behind the mirror; it strikes normally and reflects back on itself. Behind the mirror, the backward extensions of these two rays intersect between the pole ($P$) and the focus ($F$). Image Characteristics: Located behind the mirror between $P$ and $F$; diminished in size; virtual and erect.
Automotive Rear-View and Surveillance Applications
Convex mirrors are used as rear-view (wing) mirrors in automobiles and surveillance mirrors in retail stores: They form erect (upright) images of objects behind the vehicle, making traffic easier to interpret than an inverted real image. Because the image is diminished, a convex mirror provides a wider field of view than a flat planar mirror of the same size, allowing drivers to view multiple lanes of traffic.
Automotive side mirrors carry the warning: "Objects in mirror are closer than they appear", because the diminished size makes vehicles appear farther away than they actually are.
The New Cartesian Sign Convention
To ensure algebraic consistency when applying the mirror formula and calculating magnification, physicists use the New Cartesian Sign Convention.
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| NEW CARTESIAN SIGN CONVENTION MATRIX |
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| + y (Height Upward: POSITIVE) |
| ^ |
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| Incident Light Direction | |
| -------------------------> | |
| | |
| - x (Distances AGAINST light: | + x (Distances WITH light: |
| NEGATIVE) | POSITIVE) |
| <------------------------------------ P ------------------------------------> |
| (POLE) |
| | |
| | |
| | |
| v |
| - y (Height Downward: NEGATIVE) |
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Rules of the New Cartesian Sign Convention
Rule 1 (Origin Datum): The Pole ($P$) of the spherical mirror is taken as the origin of the coordinate system. All optical distances are measured along the principal axis from the pole ($P$).
Rule 2 (Horizontal Distances): Distances measured in the direction of the incident light (to the right of the pole, along the positive $x$-axis) are taken as positive ($+$). Distances measured opposite to the direction of the incident light (to the left of the pole, along the negative $x$-axis) are taken as negative ($-$).
Rule 3 (Object Distance): Because the object is always placed in front of the reflecting surface (to the left of the mirror), light travels from left to right. Therefore, the object distance ($u$) is always negative in all standard calculations:
$$u < 0 \quad (\text{Always Negative})$$
Rule 4 (Vertical Heights): Heights measured perpendicular to and above the principal axis (upward along the positive $y$-axis) are taken as positive ($+$). Erect objects and virtual images have positive heights ($h > 0, h' > 0$). Heights measured perpendicular to and below the principal axis (downward along the negative $y$-axis) are taken as negative ($-$). Inverted real images have negative heights ($h' < 0$).
Rule 5 (Focal Length Signs): The focus of a concave mirror lies in front of the mirror (to the left of the pole); therefore, the focal length and radius of curvature of a concave mirror are always negative:
$$f_{\text{concave}} < 0, \quad R_{\text{concave}} < 0$$
The focus of a convex mirror lies behind the mirror (to the right of the pole); therefore, the focal length and radius of curvature of a convex mirror are always positive:
$$f_{\text{convex}} > 0, \quad R_{\text{convex}} > 0$$
Chapter Exercises
Exercise 1: An object is placed at a distance of $15\text{ cm}$ in front of a concave mirror of focal length $10\text{ cm}$. Draw a ray diagram showing image formation. Using the New Cartesian Sign Convention, determine whether the object distance, image distance, and focal length are positive or negative, and describe the nature, position, and relative size of the image.
Exercise 2: Explain why a dentist uses a small concave mirror rather than a convex or plane mirror to inspect a patient's molars, specifying where the mirror must be positioned relative to the tooth and describing the characteristics of the resulting image.
Exercise 3: A driver observes two vehicles behind their car: Vehicle A is viewed in a planar rearview mirror, and Vehicle B is viewed in a convex side mirror. Contrast the images formed by the two mirrors in terms of field of view, orientation, image size, and perceived distance.
Solutions and Explanations
Solution 1: The object is placed at $u = 15\text{ cm}$ in front of a concave mirror of focal length $f = 10\text{ cm}$. The center of curvature is at $R = 2f = 20\text{ cm}$. Because $u = 15\text{ cm}$, the object is positioned between the focus ($F$, $10\text{ cm}$) and the center of curvature ($C$, $20\text{ cm}$). To construct the ray diagram: draw Ray 1 from the top of the object parallel to the principal axis; upon reflection, it passes through the focus ($F$); draw Ray 2 passing through the focus ($F$); upon reflection, it emerges parallel to the principal axis. The two reflected rays intersect beyond the center of curvature ($C$, beyond $20\text{ cm}$). Under the New Cartesian Sign Convention: the object distance is negative ($u = -15\text{ cm}$); the focal length of a concave mirror is negative ($f = -10\text{ cm}$); and because the image is formed in front of the mirror (to the left of the pole), the image distance is negative ($v < 0$, calculated as $v = -30\text{ cm}$). The image is located beyond $C$ at $30\text{ cm}$ in front of the mirror; it is magnified (enlarged); and it is a real and inverted image.
Solution 2: A dentist uses a small concave mirror because a concave mirror can produce an erect, virtual, and magnified image of an object when that object is placed within its focal length (between the pole $P$ and the principal focus $F$, Case 6). The dentist positions the mirror close to the patient's tooth so that the distance between the tooth and the mirror is less than the mirror's focal length ($u < f$). In this position, the reflected rays diverge in front of the mirror and appear to intersect behind the mirror, forming an upright, enlarged, virtual image of the tooth. This magnification allows the dentist to identify small cavities, cracks, and calculus deposits. A plane mirror would produce an image of the same size as the tooth without magnification, while a convex mirror would produce a diminished (smaller) image, which would be counterproductive for clinical inspection.
Solution 3: The planar interior rearview mirror and the convex exterior side mirror produce different images: For Vehicle A viewed in the plane mirror: the image is virtual, erect, and laterally inverted, with a magnification of $m = +1$ (the image is the exact same size as the actual vehicle). The image distance behind the mirror equals the object distance in front ($v = u$), providing an accurate representation of the vehicle's true distance, but offering a narrower field of view limited by the physical dimensions of the mirror glass. For Vehicle B viewed in the convex side mirror: the image is virtual, erect, and diminished ($m < 1$, smaller than the actual vehicle). Because the mirror surface curves outward, reflected rays diverge over a wider angle, providing a broader field of view that covers adjacent traffic lanes and blind spots. However, because the image is diminished, the human visual system interprets smaller retinal images as being farther away, making Vehicle B appear farther behind the car than it actually is.