  Closure phase is the phase of a closed-loop product of correlations in a $\ge
3$-element interferometer array. Its invariance to element-based phase
corruption makes it invaluable for interferometric applications that otherwise
require high-accuracy phase calibration. However, its understanding has
remained mainly mathematical and limited to the aperture plane (Fourier dual of
image plane). Here, we lay the foundations for a geometrical insight. we show
that closure phase and its invariance to element-based corruption and to
translation are intricately related to the conserved properties (shape,
orientation, and size, or SOS) of the principal triangle enclosed by the three
fringes formed by a closed triad of array elements, which is referred herein as
the "SOS conservation principle". When element-based amplitude calibration is
not needed, as is typical in optical interferometry, the 3-element interference
image formed from phase-uncalibrated correlations is a true and uncorrupted
representation of the source object's morphology, except for a possible shift.
Based on this SOS conservation principle, we present two geometric methods to
measure the closure phase directly from a 3-element interference image (without
requiring an aperture-plane view): (i) the closure phase is directly measurable
from any one of the triangle's heights, and (ii) the squared closure phase is
proportional to the product of the areas enclosed by the triad of array
elements and the principal triangle in the aperture and image planes,
respectively. We validate this geometric understanding across a wide range
range of interferometric conditions using data from the Very Large Array and
the Event Horizon Telescope. This geometric insight can be potentially valuable
to other interferometric applications such as optical interferometry. These
geometric relationships are generalised for an $N$-element interferometer.
