We present a simple yet effective end-to-end trainable deep network with geometry-inspired convolutional operators for detecting vanishing points in images. Traditional convolutional neural networks rely on aggregating edge features and do not have mechanisms to directly exploit the geometric properties of vanishing points as the intersections of parallel lines. In this work, we identify a canonical conic space in which the neural network can effectively compute the global geometric information of vanishing points locally, and we propose a novel operator named conic convolution that can be implemented as regular convolutions in this space. This new operator explicitly enforces feature extractions and aggregations along the structural lines and yet has the same number of parameters as the regular 2D convolution. Our extensive experiments on both synthetic and real-world datasets show that the proposed operator significantly improves the performance of vanishing point detection over traditional methods.