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We make the following contributions: (i) we show that the homography matrix can be parameterised with only four parameters that specify the horizon line and the vertical vanishing point, or only two if the field of view or focal length is known; (ii) We introduce a novel representation for the geometry of a line or point (which can be at infinity) that is suitable for regression with a convolutional neural network (CNN); (iii) We introduce a large synthetic image dataset with ground truth for the orthogonal vanishing points, that can be used for training a CNN to predict these geometric entities; and finally (iv) We achieve state-of-the-art results on horizon detection, with 74.52% AUC on the Horizon Lines in the Wild dataset. 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