Concave vs Convex

This entry was posted on by (updated on )
Concave vs Convex
Concave and convex describe the shape of a curve. Remember that “concave” resembles a cave!

The terms “concave” and “convex” describe the curvature of objects or mathematical functions. They’re ubiquitous in a range of disciplines, including optics, mathematics, engineering, and everyday life. Here are the definitions of concave and convex, everyday examples, and how to remember which is which.

Concave vs Convex: Definitions and Everyday Examples

Concave

Definition: An object or a function is concave if it curves inward. In simple terms, it’s hollow or bowed in, much like a cave.

Everyday Examples:

  1. A bowl.
  2. A satellite dish.
  3. A spoon’s interior.
  4. Skateboard ramps.
  5. A pie with a slice taken out of it.

Convex

Definition: An object or function is convex if it curves outward, or in other words, bulges out.

Everyday Examples:

  1. An eye
  2. A speed bump.
  3. A magnifying glass.
  4. A globe.
  5. A triangle.

Ways to Remember

  • Concave: Think “Con-cave”; it has a “cave” or an inward dip.
  • Convex: Think of “Con-vex” as “convicts” trying to escape, bulging outwards.

Concave and Convex Lenses

Concave lenses focus light inside the curve of the lens, while convex lenses focus light using the outer curve. A lens that curves like a “C” (no flat side) is both concave or convex, depending on which side of the lens you view from.

Concave Lenses

Examples: Eyeglasses for nearsighted people, cameras, telescopic sights.

Properties:

  1. Diverge light rays.
  2. Produce a virtual, diminished image.

Ways to Remember: A concave lens is “caving in”, so it spreads light away or diverges it.

Convex Lenses

Examples: Magnifying glasses, eyeglasses for farsighted people, microscopes.

Properties:

  1. Converge light rays.
  2. Produce both real and virtual images.

Ways to Remember: Convex lenses “converge” light, making things appear larger or closer.

Concave and Convex Mirrors

Concave Mirrors

Examples: Shaving mirrors, car headlights, astronomical telescopes.

Properties:

  1. Converge light rays.
  2. Can produce both real and virtual images.

Ways to Remember: Concave mirrors “converge” light, much like they’re collecting it into a focus.

Convex Mirrors

Examples: Security mirrors in stores, car side mirrors, rear-view mirrors.

Properties:

  1. Diverge light rays.
  2. Produce only virtual, diminished images.

Ways to Remember: Convex mirrors “diverge” light, spreading it out to give a wider field of view.

Concave and Convex Polygons

Concave Polygons

Examples: Star-shaped figures, snowflakes, some irregular polygons.

Properties:

  1. Have at least one interior angle greater than 180°.
  2. Have vertices that “cave” inward.

Ways to Remember: “Concave” has a “cave” or an inward dip; you can imagine it as a polygon that’s been “pinched” inward.

Convex Polygons

Examples: Squares, rectangles, triangles, regular hexagons.

Properties:

  1. All interior angles are less than 180°.
  2. No vertices “cave” inward.

Ways to Remember: Convex polygons “bulge” outward, with no indents or “caves.”

Concave and Convex Functions

Concave Functions

Examples: Exponential functions, quadratic functions with a positive leading coefficient.

Properties:

  1. Slope increases as you move along the function.
  2. Holds the property f(tx+(1−t)y) ≥ tf(x) + (1−t)f(y) for 0 ≤t ≤1.

Ways to Remember: The shape of a concave function resembles the entrance to a cave or a hill.

Convex Functions

Examples: Logarithmic functions, negative exponential functions.

Properties:

  1. Slope decreases as you move along the function.
  2. Holds the property f(tx + (1−t)y) ≤ tf(x) + (1−t)f(y) for 0 ≤t ≤1.

Ways to Remember: A convex function looks like a valley.

By understanding the basic differences and applications of concave and convex lenses, mirrors, polygons, and functions, you gain a deeper insight into how these concepts shape our world, from the glasses we wear to the roads we drive on.

References

  • Hass, Joel; Heil, C.; Weir, M. (2017). Thomas’ Calculus (14th ed.). Pearson. ISBN 978-0-13-443898-6.
  • Hecht, Eugene (2002). Optics (4th ed.). Addison Wesley. ISBN 978-0-321-18878-6.
  • Nayak, Sanjay K.; Bhuvana, K.P. (2012). Engineering Physics. New Delhi: Tata McGraw-Hill Education. ISBN 9781259006449.
  • Sines, George; Sakellarakis, Yannis A. (1987). “Lenses in antiquity”. American Journal of Archaeology. 91 (2): 191–196. doi:10.2307/505216