Thin lens and mirror equation for image distance, object distance or focal length, with magnification; Snell's law refraction and critical angle.
更新于 已验证的示例：8
With f = 10 cm and the object 30 cm away, the image forms 15 cm beyond the lens: real, inverted, reduced (m = −0.5).
Real is positive: distances to real objects and images are positive, virtual ones negative; converging lenses and concave mirrors have f > 0. For a lens a real image forms on the far side.
The thin-lens equation, 1/f = 1/dₒ + 1/dᵢ, links the focal length f, the object distance dₒ and the image distance dᵢ, and the same equation holds for spherical mirrors. Give any two and the calculator solves the third, then finds the magnification m = −dᵢ/dₒ, the image height, whether the image is real or virtual, and the power in dioptres (1/f with f in metres). Snell's law, n₁ sin θ₁ = n₂ sin θ₂, gives the refracted angle at a boundary and the critical angle beyond which light is totally reflected.
With the defaults, a converging lens of f = 10 cm and an object 30 cm away, the image forms 15 cm behind the lens, inverted and half size (m = −0.5), the arrangement in a camera. Light passing from air into water at 45° bends to 32.05°.
Signs follow the real-is-positive convention: virtual images have negative dᵢ, and diverging lenses and convex mirrors have negative f. Lenses are thin, so thickness and aberrations are ignored.
核验来源：Python 3.8 fractions: 1/dᵢ = 1/10 − 1/30 ⇒ dᵢ = 15 cm, m = −½; P = 1/0.1 m
核验来源：Python 3.8 fractions: 1/dᵢ = 1/10 − 1/5 = −1/10 ⇒ dᵢ = −10 cm
核验来源：Python 3.8 fractions: 1/dᵢ = −1/10 − 1/20 = −3/20 ⇒ dᵢ = −20/3 cm
核验来源：Python 3.8 fractions: 1/dᵢ = 1/10 − 1/20 ⇒ dᵢ = 20 cm, m = −1 (OpenStax UP3 §2.3)
1/f = 1/dₒ + 1/dᵢ, where f is the focal length, dₒ the object distance and dᵢ the image distance, all measured from the lens. For f = 10 cm and an object at 30 cm, 1/dᵢ = 1/10 − 1/30, so dᵢ = 15 cm. A negative dᵢ means a virtual image on the same side as the object, as seen through a magnifying glass.
m = −dᵢ/dₒ, and the image height is m times the object height. A negative m means the image is inverted; |m| above 1 means it is enlarged. An object 5 cm from a 10 cm magnifying glass gives dᵢ = −10 cm and m = +2, an upright virtual image twice the size. At dₒ = 2f the image is real, inverted and the same size (m = −1).
θc = arcsin(n₂/n₁), which exists only when light heads into a medium with a lower index (n₁ > n₂). Into air it is 41.8° from glass (n = 1.5), 48.6° from water (1.333) and 24.4° from diamond (2.417), which is why cut diamonds sparkle. A ray meeting the surface at more than θc from the normal is reflected completely, the principle behind optical fibre.
n₁ sin θ₁ = n₂ sin θ₂: the refractive index times the sine of the angle from the normal is the same on both sides of a boundary. Light entering water (n = 1.333) from air at 45° continues at 32.0°, bent towards the normal because water has the higher index. Going the other way the ray bends away from the normal, and beyond the critical angle it cannot leave at all.
Power P = 1/f with f in metres, measured in dioptres (D). A converging lens with f = 10 cm has P = +10 D, and a diverging lens with f = −50 cm has −2 D. Spectacle prescriptions use this unit: short sight is corrected with negative (diverging) lenses and long sight with positive ones, and the powers of thin lenses in contact add.
准确性取决于输入值和方法的假设。十进制运算使用50位有效数字，但估算、数值方法和源数据的精度可能较低；显示时的舍入并不能消除这些限制。 已按独立来源核验的计算示例：8。 例如，“Converging lens f = 10 cm, object at 30 cm”根据Python 3.8 fractions: 1/dᵢ = 1/10 − 1/30 ⇒ dᵢ = 15 cm, m = −½; P = 1/0.1 m进行核验。
OpenStax University Physics Volume 3, §2.3 Spherical mirrors and §2.4 Thin lenses; OpenStax University Physics Volume 3, §1.4 Refraction and §1.5 Total internal reflection; Hecht, Optics (5th ed.), Table 4.1 — refractive indices.
此计算器包含 8 个已解示例，答案来自独立来源。这些示例会在测试套件中运行，你也可以在此运行验证。
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