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Linear interpolation
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See: `wikipedia on linear interpolation <linear interpolation>`_.

Let us say that we have two known points $x_1, y_1$ and $x_2, y_2$.

Now we want to estimate what $y$ value we would get for some $x$ value that is
between $x_1$ and $x_2$.  Call this $y$ value estimate |--| an *interpolated*
value.

Two simple methods for choosing $y$ come to mind.  The first is see whether
$x$ is closer to $x_1$ or to $x_2$.  If $x$ is closer to $x_1$ then we use
$y_1$ as the estimate, otherwise we use $y_2$.  This is called *nearest
neighbor* interpolation.

The second is to draw a straight line between $x_1, y_1$ and $x_2, y_2$. We
look to see the $y$ value on the line for our chosen $x$.  This is *linear
interpolation*.

It is `possible to show <https://vimeo.com/124728992>`_ that the formula of
the line between $x_1, y_1$ and $x_2, y_2$ is:

.. math::

    y = y_1 + (x-x_1)\frac{y_2-y_1}{x_2-x_1}

.. nbplot::
    :include-source: false

    x1, y1, x2, y2 = 5, 6, 7, 7.5
    dx, dy = x2 - x1, y2 - y1
    x = x1 + dx * 0.6
    y = y1 + (x-x1) * dy / dx
    # Make subplots for diagram and text
    fig, d_ax = plt.subplots(1, 1, figsize=(10, 4))
    d_ax.plot([x1, x2], [y1, y2], 'o-')
    d_ax.annotate('$(x_1, y_1)$', (x1-0.2, y1-0.3), fontsize=16)
    d_ax.annotate('$(x_2, y_2)$', (x2, y2+0.2), fontsize=16)
    d_ax.annotate(
        '', xy=(x1, y1), xycoords='data',
        xytext=(x2, y1), textcoords='data',
        arrowprops={'arrowstyle': '<->', 'color': 'r'})
    d_ax.annotate(
        '', xy=(x2, y1), xycoords='data',
        xytext=(x2, y2), textcoords='data',
        arrowprops={'arrowstyle': '<->', 'color': 'k'})
    d_ax.annotate(
        '', xy=(x1, y1+0.1), xycoords='data',
        xytext=(x, y1+0.1), textcoords='data',
        arrowprops={'arrowstyle': '<->', 'color': 'g'})
    d_ax.annotate('$x_2-x_1$', (x1 + dx / 2 + 0.4, y1-0.2), fontsize=16)
    d_ax.annotate('$y_2-y_1$', (x2 + 0.1, y1 + dy / 2), fontsize=16)
    d_ax.annotate('$x-x1$', (x1 + 0.6, y1 + 0.2), fontsize=16)
    d_ax.annotate('$x$', (x + 0.1, y1 - 1), fontsize=16)
    d_ax.annotate('$y$', (x1, y + 0.1), fontsize=16)
    d_ax.axis((4.3, 7.3, 4, 8))
    lx, hx, ly, hy = d_ax.axis()
    d_ax.plot([x, x], [ly, y], 'k:')  # line in x
    d_ax.plot([lx, x], [y, y], 'k:')  # line in y
    # d_ax.axis('off')
    d_ax.annotate(r'slope : $\frac{y_2-y_1}{x_2-x_1}$', (4.5, y1-1.2), fontsize=20)
    d_ax.annotate(r'$y = y1 + (x-x_1)\frac{y_2-y_1}{x_2-x_1}$', (4.5, y2),
                  fontsize=20)
    plt.setp(d_ax.get_yticklabels(), visible=False)
    d_ax.yaxis.set_tick_params(size=0)
    plt.setp(d_ax.get_xticklabels(), visible=False)
    d_ax.xaxis.set_tick_params(size=0)
    # Hide the right and top spines
    d_ax.spines['right'].set_visible(False)
    d_ax.spines['top'].set_visible(False)

.. include:: links_names.inc